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Geodesics on the Hyperbolic Plane

Continuing from the last post with the upper half plane model as our model of the hyperbolic plane. First, it is to be noted that the fractional linear transformations or more commonly called the Moebius transformations maps the upper half plane H to itself.

Consider the transformed point under T_g on z\in H where g=\begin{pmatrix} a&b\\c&d \end{pmatrix} :

w\equiv T_g(z)=\cfrac{az+b}{cz+d}=\cfrac{ac\vert z\vert^2 + adz+bc\bar{z}+bd}{\vert cz+d\vert^2} .

The imaginary part of the transformed point is

\textrm{Im}(w)=\cfrac{w-\bar{w}}{2i}=\cfrac{(ad-bc)(z-\bar{z})}{2i\vert cz+d\vert^2}=\cfrac{\textrm{Im}(z)}{\vert cz+d\vert^2} .

Note that if \textrm{Im}(z)>0 then the above shows that \textrm{Im}(w)>0 also, and hence the earlier statetment of the map onto H itself.

To continue studying the geometry of the hyperbolic plane, one must have a notion of hyperbolic distance and this is obtained from the previous hyperbolic metric on the upper half plane. One thus defined the infinitesimal element of the hyperbolic distance to be

ds=\cfrac{\sqrt{dx^2+dy^2}}{y} .

Given a curve \gamma(t)=\{v(t)=x(t)+iy(t)\vert t\in H\}, its length is thus given by

h(\gamma)=\displaystyle\int\limits_0^1 \cfrac{\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}}{y(t)}\ dt .

For the hyperbolic distance between z,w\in H, one takes the minimum value of the length of curves connectig the two points i.e.

\rho(z,w)=\inf_\gamma h(\gamma) .

Later, it will be convenient for us to know exactly what are the shortest hyperbolic curves connecting two point on H i.e. the geodesics. For now, we note that the Moebius transformations themselves are isometries (preserve lengths). Consider the transformed points on a curve \gamma(t) :

w(t) = T(z) = \cfrac{az+b}{cz+d} = u(t) + iv(t) .

Its derivative is

\cfrac{dw}{dz}=\cfrac{a(cz+d)-c(az+b)}{(cz+d)^2}=\cfrac{1}{(cz+d)^2} .

Noting that v=y/\vert cz+d\vert^2=\textrm{Im}(w), we can write the derivative as \vert\frac{dw}{dz}\vert = v/y. Thus under the transformation, the hyperbolic distance stays invariant as shown below:

\begin{matrix} h(T(\gamma))&=\displaystyle\int\limits_0^1\cfrac{\vert\frac{dw}{dt}\vert\ dt}{v(t)}\\ &=\displaystyle\int\limits_0^1 \cfrac{\vert\frac{dv}{dz}\vert\vert\frac{dz}{dt}\vert\ dt}{v(t)} \\ &=\displaystyle\int\limits_0^1\cfrac{\vert\frac{dz}{dt}\vert dt}{y(t)}= h(\gamma) \end{matrix} .

Next, we claim that geodesics in H are either (arcs of) semicircles and vertical lines orthogonal to the real axis. We calculate first, the formula of the hyperbolic length of two points on a semicircle with centre (c,0) (see diagram below).

geodesic_arc

The two points can be coordinatized by the angles \alpha,\beta or parametrized by the following:

x=c+r\cos t\ ;\quad y=r\sin t .

Their differentials are

dx=-r\sin t dt\ ;\quad dy= r\cos t dt .

Plugging into the distance formula on a hyperbolic curve gives

\rho(\alpha,\beta)=\displaystyle\int\limits_\alpha^\beta \cfrac{\sqrt{(-r\sin t dt)^2+(r\cos t dt)^2}}{r\sin t}= \displaystyle\int\limits_\alpha^\beta \cfrac{r dt}{r\sin t}

Note the denominator of the final integrand can be rexpressed as 2\sin\frac{t}{2}\cos\frac{t}{2} and observing that

d(\ln\tan\frac{t}{2}) = \cfrac{1}{\tan\frac{t}{2}}\ \cfrac{\sec^2\frac{t}{2}}{2}\ dt ,

one obtains

\rho(\alpha,\beta)=\displaystyle\int\limits_\alpha^\beta d\left(\ln\tan\frac{t}{2}\right)=\ln\left(\cfrac{\tan\beta/2}{\tan\alpha/2}\right)

Using trigonometric identity,

\csc\theta-\cot\theta=\cfrac{1}{\sin\theta}-\cfrac{\cos\theta}{\sin\theta}=\cfrac{1-\cos^2\theta/2+\sin^2\theta/2}{2\sin\theta/2\cos\theta/2}=\cfrac{\sin\theta/2}{\cos\theta/2} ,

we can reexpress the distance (following Stahl) as

\rho(\alpha,\beta)=\ln\left(\cfrac{\csc\beta-\cot\beta}{\csc\alpha-\cot\alpha}\right) .

For the case of two points on the vertical line

geodesic_vert

the distance formula (with dx=0) gives

\rho(y_1,y_2)=\displaystyle\int\limits_{y_1}^{y_1} \cfrac{dy}{y}=\ln\left(\cfrac{y_2}{y_1}\right) .

Next, we show the distance formula given by the (claimed) geodesics indeed give the infimum distance. Consider an arbitrary curve between P and Q (not collinear vertically):

nongeodesic1

Using the same coordinatization x=d+r\cos\theta,\ y=r\sin\theta but with r depend on \theta, we obtain

\cfrac{dx}{d\theta}=r'\cos\theta-r\sin\theta\ ;\quad\cfrac{dy}{d\theta}=r'\sin\theta+r\cos\theta .

Hence,

dx^2+dy^2=\left(\cfrac{dx}{d\theta}\right)^2d\theta^2+\left(\cfrac{dy}{d\theta}\right)^2d\theta^2=(r'^2+r^2)d\theta^2 .

Thus, the distance measured through arbitrary curve connecting P and Q obeys

\displaystyle\int\limits_\alpha^\beta \cfrac{\sqrt{r'^2+r^2}}{r\sin\theta}\ d\theta\geq\displaystyle\int\limits_\alpha^\beta \cfrac{\sqrt{r^2}}{r\sin\theta} d\theta=\int_\alpha^\beta\csc\theta d\theta=\rho(\alpha,\beta) .

This shows the arc of the semicircle connecting P and Q gives the infimum hyperbolic distance. Next we do the same for arbitrary curves joining two vertically collinear points.

nongeodesic2

Let x be a function of y i.e. x=x(y). Then

\displaystyle\int\limits_{y_1}^{y_2} \cfrac{\sqrt{dx^2+dy^2}}{y}=\displaystyle\int\limits_{y_1}^{y_2} \cfrac{\sqrt{x'^2dy^2+dy^2}}{y}\geq\displaystyle\int\limits_{y_1}^{y_2} \cfrac{dy}{y}=\ln\cfrac{y_2}{y_1} .

Again, the vertical line joining the two points give the least distance in comparison to other curves joining them, hence the earlier-claimed geodesics.

Finally we would like to give the solution to Exercise 4 of Katok’s article where one could essentially transform any semicircle (geodesic) to the vertical line and hence making any other proofs concerning geodesics easier. Let the semicircle be centred at the origin with radius \alpha and the vertical line be the y-axis. Both can be parametrized as z=\alpha e^{i\theta}\ ,\ z=i\tan\theta/2 respectively. Consider then the transformation T(z)=-(z-\alpha)^{-1}+\beta. Plugging in the parametrization gives

T(z)=\cfrac{1}{\alpha(1-e^{i\theta})}+\beta=\cfrac{1+\beta\alpha(1-e^{i\theta})}{\alpha e^{i\theta/2}(e^{-i\theta/2}-e^{i\theta/2})} .

Putting \beta\alpha = -1/2 and multiplying top and bottom by e^{-i\theta/2} gives

T(z)=-\cfrac{1}{2\alpha}\ \cfrac{e^{-i\theta/2}+e^{i\theta/2}}{e^{i\theta/2}-e^{-i\theta/2}} .

Note that the second factor is already the reciprocal of i\tan\theta/2. Thus by putting \alpha=1/2 and composing the transformation with an inversion S(z)=-1/z gives the necessary transformation i.e.

(S\circ T)(z)=\left(\cfrac{1}{z-\frac{1}{2}} + 1\right)^{-1} .

References

  1. S. Katok, “Fuchsian Groups, Geodesic Flows on Surfaces of Constant Negative Curvature and Symbolic Coding of Geodesics”, (http://www.math.psu.edu/katok_s/cmi.pdf)
  2. R. Hayter, “The Hyperbolic Plane – A Strange New Universe” , (http://www.maths.dur.ac.uk/Ug/projects/highlights/CM3/Hayter_Hyperbolic_report.pdf)
  3. S. Stahl, A Gateway to Modern Geometry: The Poincare Half-Plane, (Jones & Bartlett, 2007).
  4. J. Hilgert, “Maass Cusp Forms on \textrm{SL}(2,\mathbb{R})“, (http://www.math.uni-paderborn.de/~hilgert/Metz05web.pdf)
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Models of Hyperbolic Plane

Resurrecting this blog. Recently I have been giving lectures on hyperbolic geometry to my students including student trainees. The lectures allow me to brush up on basics of hyperbolic geometry for upcoming conference and also to help the trainees do their mini projects. I begin with the models of hyperbolic plane.

Hyperboloid Model

Everyone knows the unit (two-) sphere whose equation is x_1^2 + x_2^2 + x_3^2 = 1:

220px-Sphere_wireframe_10deg_6r_svg

If one changes the sign in one variable of the sphere-equation, x_1^2 + x_2^2 - x_3^2 = 1, one obtains the one-sheeted hyperboloid:

150px-HyperboloidOfOneSheet

If one changes the sign of the constant on the right hand side, x_1^2 + x_2^2 - x_3^2 = -1, one gets the two-sheeted hyperboloid:

150px-HyperboloidOfTwoSheets

These disconnected pieces remind us the hyperbola curves we plotted in schools x^2 - y^2 =\pm 1:

300px-Drini-conjugatehyperbolas_svg

One should also mention the other possibility from the two-forms of the hyperboloid earlier, the one with zero on the right hand side, giving a double cone reminiscent of the light cones in relativity.

160px-DoubleCone

These null surfaces are the asymptotes of the hyperboloids (very much like the asymptotes of the hyperbola).

Returning to the double-sheeted hyperboloid; let us restrict to the positive sheet namely

L=\{(x_1,x_2,x_3)\in\mathbb{R}^3\vert x_1^2+x_2^2-x_3^2=-1,\ x_3>0\} ,

it gives the hyperboloid model of the hyperbolic plane. The surface can be parametrised in ‘angular’ coordinates much in the same way as the sphere via

x_1=\sinh\theta\cos\phi;\ x_2=\sinh\theta\sin\phi;\ x_3=\cosh\theta,

which leads to x_1^2+x_2^2-x_3^2=\sinh^2\theta-\cosh^2\theta= -1. Note the use of hyperbolic functions. Note also the form of bilinear form one can associate to \mathbb{R}^3 to form the equation of the hypwerboloid i.e.

(x,y)_{2,1} = x_1y_1 + x_2y_2 - x_3y_3\

Using this bilinear form, one could actually form the metric on L by forming the product (dx,dx)_{2,1} to give

ds^2 = dx_1^2+dx_2^2-dx_3^2\ .

Beltrami-Klein Model

In the hyperboloid model L above, it is tangent to the plane K of x_3=1. Next we project points on L to this plane by drawing a line to the origin (see Fig below).

BKModel

The point l=(x_1,x_2,x_3)\in L is projected to the point k=(\eta_1,\eta_2)\in K. By using similar triangles, \eta_i=\eta_i/1=x_i/x_3 for i=1,2. Note that

\eta_1^2+\eta_2^2=\cfrac{x_1^2+x_2^2}{x_3^2}=\cfrac{x_3^2-1}{x_3^2}=1-\frac{1}{x_3^2}<1.

Thus under this project the hyperboloid L is mapped to a unit disk K, which is the Beltrami-Klein disk model of the hyperbolic plane. By using the inverse coordinate transformation

x_i=\cfrac{\eta_i}{\sqrt{1-\eta_1^2-\eta_2^2}}\ ,\ (i=1,2);\quad x_3=\cfrac{1}{\sqrt{1-\eta_1^2-\eta_2^2}},

one can get the hyperboloid differentials in terms of these new coordinates as

dx_i=\cfrac{(1-\eta_1^2-\eta_2^2)d\eta_i+\eta_i(\eta_1 d\eta_1+\eta_2 d\eta_2)}{(1-\eta_1^2-\eta_2^2)^{3/2}}

for i=1,2\ and

dx_3=\cfrac{(\eta_1 d\eta_1+\eta_2 d\eta_2)}{(1-\eta_1^2-\eta_2^2)}.

Hence the metric for the B-K disk to be

ds^2=\cfrac{d\eta_1^2+d\eta_2^2}{1-\eta_1^2-\eta_2^2} + \cfrac{(\eta_1d\eta_1+\eta_2d\eta_2)^2}{(1-\eta_1^2-\eta_2^2)^2}.

Hemispherical Model

Another interesting model is to project the points of hyperboloid further to a hemisphere J (would aid us later to get other more convenient models).

hemispheremodel

The point l is now projected down to j=(\eta_1,\eta_2,\eta_3) from k of B-K model where the additional coordinate is given by \eta_3=1/x_3,\ x_3>0. It is easy to show that these coordinates now form the equation of a (hemi-)sphere

\eta_1^2+\eta_2^2+\eta_3^2=1.

Hence, the hemisphere model of the hyperbolic plane. Note that this (hemi-)sphere has a different metric inherited from the hyperboloid i.e.

ds^2=\cfrac{d\eta_1^2+d\eta_2^2+d\eta_3^2}{\eta_3^2}.

Poincare Disk Model

Using the hemisphere model, we can now make a stereographic projection from J to the plane \eta_3=0. The point j=(\eta_1,\eta_2,\eta_3) gets mapped to i=(\xi_1,\xi_2) on the Poincare disk I.

PoincareDiskModel

The relation between these coordinates are given by the similar triangle relations

\cfrac{\xi_1}{\eta_1}=\cfrac{\xi_2}{\eta_2}=\cfrac{1}{1-\eta_3}.

One can work backwards (easier) to show that the Poincare disk metric

ds^2= \cfrac{4(d\xi_1^2+d\xi_2^2)}{(1-\xi_1^2-\xi_2^2)}

is equivalent to the metric of the hemisphere model earlier.

Upper Half-Plane Model

Now in the Poincare disk, we use the stereographic projection from (0,0,-1) to \eta_3=0 plane. We could use a different projection namely from (0,-1,0) on the hemisphere J to the plane \eta_2=0. The point j= (\eta_1,\eta_2,\eta_3) gets mapped to h=(x,y) on the plane mentioned.

uhpmodel

The relation again between these coordinates are given by the similar triangles:

\cfrac{x}{\eta_1}=\cfrac{y}{\eta_3}=\cfrac{1}{1-\eta_2}.

Unlike the earlier stereographic projection that squashes the hemisphere into a bounded (Poincare) disk, the present projection gives an unbounded plane due to the point (0,-1,0) gets mapped to infinity. Note also that \eta_3>0 implies that y>0. The metric can also be worked (as in the previous case) to

ds^2= \cfrac{dx^2+dy^2}{y^2}.

This upper half plane can be realised as

H=\{z\in\mathbb{C}\vert\textrm{Im}(z)>0\}.

This is probably one of the most convenient model to work with. It can be realised as a homogeneous space. Consider the action of \textrm{SL}(2,\mathbb{R}) on H given by the fractional linear transformation

\begin{pmatrix} a&b\\c&d \end{pmatrix} \cdot z \mapsto \cfrac{az+b}{cz+d}\equiv z'

where a,b,c,d\in\mathbb{R};\ ad-bc=1. Note the action of \begin{pmatrix} -a&-b\\-c&-d \end{pmatrix} gives the same effect. Thus the effective group is really \textrm{PSL}(2,\mathbb{R}) = \textrm{SL}(2,\mathbb{R})/\pm I. The group action could also be conveniently put into matrix multiplication form in the following way:

\begin{pmatrix} a&b\\c&d \end{pmatrix}\cdot \begin{pmatrix} z\\1 \end{pmatrix} = \begin{pmatrix} az+b\\cz+d \end{pmatrix} \equiv \begin{pmatrix} \frac{az+b}{cz+d}\\1 \end{pmatrix}.

It would seem like that each point on H can be made in correspondence with each element of \textrm{PSL}(2,\mathbb{R}) but one ought to check if any of the group elements keep the point fixed. Consider for example the subgroup \textrm{SO}(2)\subset\textrm{SL}(2,\mathbb{R}). Elements \begin{pmatrix} \cos\theta&-\sin\theta\\ \sin\theta&\cos\theta \end{pmatrix}\in\textrm{SO}(2) keeps z=i fixed:

\begin{pmatrix} \cos\theta&-\sin\theta\\ \sin\theta&\cos\theta \end{pmatrix}\ \begin{pmatrix} i\\1 \end{pmatrix}= \begin{pmatrix} i(i\sin\theta+\cos\theta)\\ i\sin\theta+\cos\theta \end{pmatrix}\equiv \begin{pmatrix} i\\1 \end{pmatrix}.

Thus in general for each point there will be effectively an (conjugated) \textrm{SO}(2) that keeps the point fixed. Thus the upper half plane can be realised as the quotient space \textrm{PSL}(2,\mathbb{R})/\textrm{PSO}(2).

In the next few forthcoming posts, we will investigate this model of the hyperbolic plane further

References

  1. Pages from http://en.wikipedia.org/wiki/Main_Page
  2. S. Katok, “Fuchsian Groups, Geodesic Flows on Surfaces of Constant Negative Curvature and Symbolic Coding of Geodesics”, (http://www.math.psu.edu/katok_s/cmi.pdf)
  3. J.W. Cannon, W.J. Floyd, R. Kenyon & W.R. Parry, “Hyperbolic Geometry” (http://www.math.brown.edu/~rkenyon/papers/cannon.pdf)
  4. R. Hayter, “The Hyperbolic Plane – A Strange New Universe” (http://www.maths.dur.ac.uk/Ug/projects/highlights/CM3/Hayter_Hyperbolic_report.pdf)
  5. J. Hilgert, “Maass Cusp Forms on \textrm{SL}(2,\mathbb{R})“, (http://www.math.uni-paderborn.de/~hilgert/Metz05web.pdf)

Making Spaces

This is a set of notes from some talk way back in February 2011 and I decided to type it out and it continues naturally from the last post on set theory.

The question posed is what is a space? More often than not, physicists tend to think that spaces are ‘places’ where objects can be contained in very much like the space(-time) we live in. But these ideas have further properties with them such as breaking them into subregions, joining regions into bigger ones, distinguishability of points and regions, all of which are captured in the mathematicians’ notion of topology.

A topology on a set X is a collection \mathcal{T} of subsets of X called open sets satisfying

  1. X and \emptyset are elements in \mathcal{T}
  2. Closure under finite intersections
    If U_1, U_2, \dotsc , U_n \in \mathcal{T}, then U_1\cap U_2 \cap \dotsc U_n \in \mathcal{T}.
  3. Closure under arbitrary unions
    If \{ U_\alpha \}_{\alpha\in A} is any collection of elements in \mathcal{T} (labelled by index set A, then \cup_{\alpha\in A} U_\alpha \in \mathcal{T}.

The pair (X, \mathcal{T}) is called a topological space. Often the case, physicists tend to refer to X as being the space itself with the understanding some standard choice of topology has been made.

Before elaborating on further ideas of topological space, we need to make known some intuitive ideas of what is an open set. First, we take the set \mathbb{R} with an ordered relation <. Intervals of \mathbb{R} form needed subsets. For example,

(a, b) = \{ x\in\mathbb{R} \vert a < x < b\} is an open interval.

[ a, b] = \{ x\in \mathbb{R} \vert a \leq x \leq b \} is a closed interval.

Note that (a, b) form the open sets while [a, b] form the closed sets. The open intervals in fact form the standard topology on \mathbb{R}.

Closed sets are different creatures but not unrelated.  A set A if topological space X is closed if set X-A is open i.e. (X-A)\in \mathcal{T}. For example, [a, \infty) is closed since (-\infty, a) is open. In fact there is some form of duality relationship between the two. One can even define a topological space using closed sets.

Let X be a topological space. Then

  1. Arbirary intersections of closed sets in X are closed.
  2. Finite union of closed sets in X are closed.

Note also the dual behaviour of the set operations of union and intersection. It is important to note that X, \emptyset are both open and closed. Taking the earlier as definition then the empty set is closed since its complement X is open;  and X is also closed since its complement \emptyset is open. A set that is both closed and open is called clopen.

To prove the first point, consider \{ A_\alpha \}_{\alpha\in J} of closed sets. By deMorgan’s law, we have

X - \cap_{\alpha\in J} A_\alpha = \cup_{\alpha\in J} (X - A_\alpha).

But (X-A_\alpha) is open and arbitrary union of open sets are open. Hence (X-\cap_{\alpha\in J} A_\alpha) is open, which then implies \cap_{\alpha\in J} A_\alpha is closed.

For the second point, consider closed sets A_i for i=1,\dotsc , n. By de Morgan’s law, we have

X - \cup_{i=1}^n A_i = \cap_{i=1}^n (X-A_i)

Since (X-A_i) is open and finite intersections of open sets are open, then (X-\cap_{i=1}^n A_i) is open or \cap_{i=1}^n A_i is closed. QED.

A set can also be neither open nor closed. Consider \mathbb{R} with the standard topology. The intervals (-\infty, a) and (a,\infty) are both open but [ a,b) is not open and it is also not closed. This can be understood in the following way; its complement being

\mathbb{R} - [a,b) = (-\infty, a) \cup [b,\infty)

is the union of an open set and a closed set. Thus the said property of [a,b).

It is important to note that the open property of a set is not really universal. We can see this by setting up a counterexample. Consider the set Y = [0,1]\cup [2,3]\subset \mathbb{R} as a topological space with the standard topology. Now [0,1] is open in Y since Y - [0,1] = [2,3] is closed. But then [0,1] is closed in \mathbb{R}; hence the open property does not carry over to the container space.

Having said much about the idea of open sets that underlies the definition of topological space, we can now see the reason why in defining (X,\mathcal{T}), we only allow for finite intersections of open sets. If this limitation is lifted, then one can consider for example, the intersection of all intervals (-\frac{1}{n}, n) of \mathbb{R}. The result would give the set \{ 0 \}, which is not open (hence failing the condition).

Let’s try to go beyond one-dimensional \mathbb{R} and discuss the analogue of open intervals for open sets. One problem is the ordering property of \mathbb{R} is no longer there. Thus one has to invoke other structures for the purpose of introducing open sets. Consider \mathbb{R}^2 as the metric space (\mathbb{R}^2, d) with standard (Euclidean) metric d. Then

D = \{ y\in\mathbb{R}^2 \vert d(x,y) < R\}

is an open disk centred at x\in \mathbb{R}^2 and this serves as the open sets for \mathbb{R}^2. Taking its closure,

\overline{D} = \{ y\in\mathbb{R}^2 \vert d(x,y) \le R\}

is a closed disk and this serves as the closed sets. It is now easy to see how this can be generalised to an open (closed) ball in (\mathbb{R}^n, d) and in higher-dimensional metric spaces. In fact, the generalisation does not stop there. One can build spaces much different from the usual familiar notion of spaces such as space of functions (important in functional analysis), spaces of lines, spaces of solutions etc. For a good read of examples of topological spaces, try this. A route to further generalisation in this case is to move from open sets to related ideas of neighbourhoods of points and to ideas of basis and filiters over which issues like convergence can be discussed. This is beyond the scope of the present post.

Let us however get more elementary for the moment and consider sets of discrete elements to demonstrate further ideas of topology. Consider the following examples over the set X=\{1, 2, 3\}.

Let \mathcal{T}_1 = \{ \emptyset, \{ 1. 2 \}, \{ 2, 3 \}, X\}. Checking on its topology property e.g.

\{ 1, 2\} \cup \{ 2, 3\} = X \in \mathcal{T}_1

which seems to be fine but the following property

\{ 1, 2 \} \cap \{ 2, 3\} = \{ 2\} \notin \mathcal{T}_1

fails. Thus \mathcal{T}_1 is not a topology on X.

If we let \mathcal{T}_2 = \{ \emptyset, \{ 1\}, \{ 2,3\}, X \}, then a check on its topological properties shows that it is a valid topology on X e.g.

\{ 1\} \cap \{ 2, 3\} = \emptyset \in \mathcal{T}_2

\{ 1\} \cup \{ 2, 3\} = X \in \mathcal{T}_2

Next, we can consider \mathcal{T}_3 = \{ \textrm{all subsets of } X\}. It can be easily checked that \mathcal{T}_3 forms a topology on X. The topology is called a discrete topology on X and (X, \mathcal{T}_3) is called a discrete space.

We could also form the topology \mathcal{T}_4 = \{ \emptyset , X \} and this is said to be the trivial topology on X. The different topologies can be depicted in the following picture.

The above example shows that one can define different topologies on a set/space. Note also that \mathcal{T}_4 \subset \mathcal{T}_2 \subset \mathcal{T}_3 and we say this as \mathcal{T}_4 is coarser than \mathcal{T}_2 and \mathcal{T}_3 is finer than \mathcal{T}_2. The finer topologies allow us to distinguish more points of the space (later we will discuss separation axioms).

Now, two different definition of topologies may not necessarily give different topological  spaces. Consider back the metric space (\mathbb{R}^2, d_E) with Euclidean metric  d_E (x,y) = \left(\sum_{i=1}^2 (x_i - y_i)^2\right)^{1/2} whose open set is the open disk. We can equip the space with a different metric, the taxicab metric, d_T (x,y) =\sum_{i=1}^2 \vert x_i - y_i \vert whose open set will be an open square. One can inscribe the square in the disk indicating the d_T-induced topology finer that the d_E-induced topology. But so can the disk be inscribed in the square indicating the reverse. Herce in this case, they are equivalent as topological spaces.

The final point for this post is how can we separate points in space. Often mentioned in theoretical physics literature, when defining a manifold, is the Hausdorff property. A topological space X is Hausdorff if given any pair of distinct points x_1, x_2 \in X, there exists neighbourhoods (open sets cointaining respective points) U_1 of x_1 and U_2 of x_2 such that U_1 \cap U_2 = \emptyset. Note the utility of open sets separating points.

There can be different degrees of variability on how we separate points. We list the different axioms of separability for reader’s reference.

T_0 axiom: If x_1, x_2 \in X, there exists open set O \in\mathcal{T}  such that
(x_1\in O \wedge x_2\notin O) \vee (x_2\in O\wedge x_1\notin O).
A T_0 space is called a Kolmogorov space.

T_1 axiom: If x_1, x_2 \in X, there exist open sets O_1, O_2 \in\mathcal{T} such that
(x_1\in O_1 \wedge x_2\in O_2) \wedge (x_2\notin O_1 \wedge x_1\notin O_2).
A T_1 space is called a Frechet space.

T_2 axiom: If x_1, x_2 \in X, there exist open sets O_1, O_2 \in\mathcal{T} such that
(x_1\in O_1 \wedge x_2 \in O_2) \wedge (O_1\cap O_2 = \emptyset).
This is the case of the Hausdorff space.

T_3 axiom: If A is a closed set with x_2\notin A, there exist open sets O_A, O_2 such that
(A\subset O_A \wedge x_2 \in O_2) \wedge (O_A \cap O_2 = \emptyset).

T_4 axiom: If A, B are disjoint closed sets in X, there exist open sets O_A, O_B such that
(A\subset O_A \wedge B\subset O_B) \wedge (O_A\cap O_B = \emptyset).

T_5 axiom: If A, B are separated sets (i.e. \bar{A} \cap B = \emptyset = A \cap \bar{B}) in X, there exist open sets O_A, O_B such that
(A\subset O_A \wedge B\subset O_B) \wedge (O_A\cap O_B = \emptyset).

Combining conditions T_1 \wedge T_2 gives what is known as regular space and combining T_1 \wedge T_4 gives a normal space. We also have the T_{2\frac{1}{2}} space where the T_2 axiom is supplemented by the condition \bar{O}_1 \cap \bar{O}_2 = \emptyset, which is called a completely Hausdorff space. Most of these differences are however ignored by theoretical physicists.

References

List of books referred to for this post.

What They Don’t Teach You About Sets 2

This is the second installment of notes of my lecture in LuFTER 1/2011. In the last post, we are limited to relationships on a given set. With so limited tools, we will not be able to do much apart from say, solving taxonomic problems. Here, we will now introduce more structures, namely operations on sets.

Set Operations

The two basic operations of set theory are the well-known set union and set intersection.

The union X\cup Y of sets X and Y is the set of objects whose members are either members of X or members of Y i.e.

X\cup Y=\{x:(x\in X)\lor(x\in Y)\} .

The set union obeys commutativity

X\cup Y=Y\cup X ,

and associativity

X\cup(Y\cup Z)=(X\cup Y)\cup Z .

The intersection X\cap Y of sets X and Y is the set of objects whose members are members of both X and Y i.e.

X\cap Y=\{x: (x\in X)\land(x\in Y) .

Similarly the set intersection also obeys commutativity and associativity:

X\cap Y=Y\cap X ;

X\cap(Y\cap Z)=(X\cap Y)\cap Z .

Exercise 1: Prove the commutativity and associativity laws for the set union and intersection.

At this juncture, one should probably have noticed the correspondence between set union and intersection with the logical or and logical and respectively. Later, we will see that these set operations do indeed give the algebra of Boolean logic. (Question: Which comes first?) It is also good to highlight one can extend both union and intersection to a family of sets X_i where i\in I for some labelling set I; namely \bigcup_{i\in I} X_i and \bigcap_{i\in I} X_i. You will probably see them in defining topological space.

One could go further to define the idea of set-theoretic difference X-Y (perhaps as opposed to the union) as

X-Y=\{x:(x\in X)\land(x\notin Y) .

Sometimes, it is also denoted as X\backslash Y. An example is the set difference \mathbb{R}-\mathbb{Q} between the set of real numbers \mathbb{R} and the set of rational numbers \mathbb{Q}; the resultant set is the set \mathbb{J} of irrational numbers.

Exercise 2: Prove X-Y=X-(X\cap Y).

Note that if X\cap Y=\emptyset, then X and Y is said to be disjoint. Related to the set difference operation is the symmetric difference i.e.

X\Delta Y=(X-Y)\cup(Y-X) ,

and this has “better” properties:

  • X\Delta Y=Y\Delta X;
  • X\Delta Y=(X\cup Y)-(Y\cap X);
  • (X\Delta Y)\Delta Z=X\Delta(Y\Delta Z).

Often one build sets from one large set (the universe of discourse) and this large set, we called universal set U. Within such set then, we can now define an operation called complement X^c of the set X\subset U:

X^c=\{x:x\in U\land x\notin X\} .

This complement operation obeys the following:

  • (X^c)^c=X;
  • \emptyset^c=U;
  • U^c=\emptyset;
  • X\cup X^c=U;
  • X\cap X^c=\emptyset.

The complement operation also obeys the deMorgan’s laws:

  • (X\cup Y)^c=X^c\cap Y^c;
  • (X\cap Y)^c=X^c\cup Y^c.

One can see that the complement really acts like a NOT gate.

Exercise 3: Prove the properties of the complement operation including deMorgan’s laws.

To complete the logic operations analogy, one adds further the distributivity law whenever one mixes both the union and intersection operations together:

  • X\cup(Y\cap Z)=(X\cup Y)\cap(X\cup Z);
  • X\cap(Y\cup Z)=(X\cap Y)\cup(X\cap Z).

Exercise 4: Prove the distributive laws.

Putting together all the operations together with the sets X,Y,Z in U, they form what is known as the Boolean algebra.

We end this part by giving an interesting example of constructing the “natural numbers” using set operations. We begin with the empty set \emptyset, which we conveniently called it 0. Next, we define what we called as the successor set S(n)=n\cup\{n\} giving the following sequence (with more relabelling):

  • 0:=\emptyset;
  • 1:=S(0)=0\cup\{0\}=\emptyset\cup\{0\}=\{0\};
  • 2:=S(1)=1\cup\{1\}=\{0\}\cup\{1\}=\{0,1\};
  • 3:=S(2)=2\cup\{2\}=\{0,1\}\cup\{2\}=\{0,1,2\};
  • \qquad\vdots

The collection \{0,1,2,3,\cdots\} forms the set \omega of natural numbers, which can be shown to obey the Peano postulates:

  • 0\in\omega;
  • \forall n\in\omega\Rightarrow S(n)\in\omega;
  • \forall n,m\in\omega\ (n\neq m)\Rightarrow(S(n)\neq S(m));
  • \forall X\subset\omega,\ ((0\in X)\land(\forall n\in X(S(n)\in X)))\Rightarrow X=\omega.

The set \omega can then be identified with the natural numbers \mathbb{N}=\{0,1,2,\cdots\}. One could easily see that for m,n\in\omega, then m\in n\Rightarrow m\subset n implying the orderedness of numbers, a property that we will touch upon later. In fact we could do more by defining addition and multiplication using set-theoretic relations and operations.

It is often remarked that this example is like creating something (numbers) out of nothing (the empty set). In some sense, it is but there is no need to philosophize it too much (and in no way it is like the creation operation!). It is simply an abstraction of the counting operation. An analogy can be made with by first peering into an empty box and later put the empty box in another box, and the latter box in one other box ad infinitum, very much like the Russian doll. If there is anything to philosophize on, is that numbers can be thought of as an abstract mental construct that we associate with counting – a fact that we often forget.

References

These are additional references since the first part:

What They Don’t Teach You About Sets 1

Resurrected this blog (again) for posting my postgraduate lectures/talks here. Here are the notes from LuFTER 1/2011 February 9, 2011. The Lunchtime Foundational Theory Expositions and Ruminations will consists of mainly lectures to my postgraduate students and occasionally research seminars, proposals etc.

The present ongoing lectures are meant to introduce mathematical structures in theoretical physics with perhaps Nirmala Prakash’s book as a guide of topics to be covered (but not necessarily adhering to it). The book itself is skewed towards topics for string theory but we will also cover other topics. The first topic will be on set theory. The mischievous title of the lecture is taking cue from Devlin’s book entitled “The Joy of Sets”, which is referenced.

Relationships and Sets

In most (conventional) physics, often use very rich mathematical structures (e.g. differential geometry) from the outset, which assume many things. Advances in foundational theories (e.g. quantum gravity) tend to relook at the usage basis of these structures and either generalize them or opt for primitive structures. So, what would be considered the most primitive structure? Set theory seems to fit the role with its common usage as a mathematical language. However, usually physicists are only exposed to very elementary ideas of set theory and perhaps not see its full power often only found in more advanced mathematics course. We hope to remedy this a little. In fact the deeper usage of set theory is really the following:

  • understanding the infinite
  • foundational subject matter of mathematics
  • common mode of reasoning

So what is a set? Its fundamental idea is simply the ability to regard a collection of objects as a single entity (the set). This sounds circular. In fact, in set theory, the undefinables are really the notion of a set and the relation “is an element of”. We introduce notation:

  • x \in X” which means “x is an element of X“;
  • x \notin X” which means “x is not an element of X“.

In forming sets, one can either

  • enumerate the elements or members of the set e.g. X = \{ x_1 , x_2 , \cdots , x_n , \cdots \}; or
  • describe by using some property P e.g. X = \{ x : P(x) \}, which is the set of all x for which P(x) holds. Example: \mathbb{C} = \{ z : z \textrm{ is a complex number} \}.

Using normal sentences to describe a set may not be best. Better, use logical statements. For this, we introduce the logical notations:

  • \Rightarrow means “implies”;
  • \Longleftrightarrow means “if and only if”;
  • \neg means “not”;
  • \wedge means “and”;
  • \vee means “or”;
  • \forall means “for all”;
  • \exists means “there exists”.

With these, one can start building logical statements for the set building. Some examples logical statement are given below (which also shows that some logical operations can be “derived” from others.

Example 1: P \Longleftrightarrow Q is the same as (P\Rightarrow Q) \land (Q\Rightarrow P)
One can build the truth table to show this is true.

\begin{array}{c|c|c} P\Rightarrow Q & Q\Rightarrow P & (P\Rightarrow Q)\land(q\Rightarrow Q)\\ \hline \textrm{False} & \textrm{False} & \textrm{False}\\ \textrm{False} & \textrm{True} & \textrm{False}\\ \textrm{True} & \textrm{False} & \textrm{False}\\ \textrm{True} & \textrm{True} & \textrm{True}\end{array}

Note that we could have build the truth table out of atomic statements instead of compound ones but we leave this for the reader to elaborate.

Exercise 1: Show that P\Rightarrow Q is the same as (\neg P)\vee Q.

Example 2: P\vee Q is the same as \neg((\neg P)\land(\neg Q)).
Now we build the truth table from the atomic statements for illustration.

\begin{array}{c|c|c|c|c|c} P & Q & \neg P & \neg Q & (\neg P)\land(\neg Q) & \neg((\neg P)\land(\neg Q))\\ \hline \textrm{False} & \textrm{False} & \textrm{True} & \textrm{True} & \textrm{True} & \textrm{False}\\ \textrm{False} & \textrm{True} & \textrm{True} & \textrm{False} & \textrm{False} & \textrm{True}\\ \textrm{True} & \textrm{False} & \textrm{False} & \textrm{True} & \textrm{False} & \textrm{True}\\ \textrm{True} & \textrm{True} & \textrm{False} & \textrm{False} & \textrm{False} & \textrm{True}\end{array}

This example is illustrative of the fact that one does not need all the logical operations; here, the “or” operation has been replaced by a combination of a “not” and an “and”. It is in fact well known that the “nand” gate (combining “not” and “and”) is a universal gate for classical computations.

In handling or consructing sets abstractly, it is important to ponder on Quine’s dictum “No entity without identity”. How would one know that two so-called abstract entities (sets) are not one and the same? Here, we state the following axiom of extensionality telling us when two sets X and Y are the same:

X=Y \iff \forall x (x\in X) \Longleftrightarrow (x\in Y) .

Thus, for example \{a,b,c\} = \{c,a,b\}. In this regard, there is one important set that we need to construct, namely the empty set \emptyset or \{\}. It can be defined as

\emptyset = \{ x: x \neq x \} .

Note from this definition, in principle we could have started with x coming from different sets, say X giving \emptyset_X and \chi giving \emptyset_\chi. But by virtue of axiom of extensionality, the empty set is unique, being the set with no elements (Exercise 2: Prove this).

Exercise 3: Using results of exercise 1, prove the statement x\in\emptyset \Rightarrow P(x) is true for all x.

An easy way to define more sets is to consider another set relation, namely subsets. We define Y is a subset of X, written as Y\subseteq X, if and only if every element of Y is an element of X i.e.

Y\subseteq X\Longleftrightarrow\forall x((x\in Y)\Rightarrow(x\in X)) .

Note that in this case X is also called superset of Y i.e. X\supseteq Y. In both these relations, it is possible that X=Y. But if you would like to consider otherwise i.e. considering Y is a proper subset of X, then we write Y\subset X where

Y\subset X\Longleftrightarrow(Y\subseteq X)\land(Y\neq X) .

Another important set concept is the idea of a power set \mathcal{P}(X) of the set X, which is the set of all subsets of X i.e.

\mathcal{P}(X) = \{ Y :Y\subseteq X\} .

We will illustrate the idea of power sets in a minute but before that one introduces another set relation namely the cardinality. Cardinality is a measure of the “size’ of the set particularly by looking into the “number” of its elements. We denote the cardinality of a set X by \sharp(X) or \textrm{card}(X).

Example 3: Consider the set X=\{\Diamond\}. Its cardinality is \sharp(X)=1. The power set of X is \mathcal{P}(X)=\{\emptyset,\{\Diamond\}\} and hence \sharp\mathcal{P}(X)=2. One could proceed further to find the power set of \mathcal{P}(X) itself:

\mathcal{P}(\mathcal{P}(X))=\{\emptyset,\{\emptyset\},\{\Diamond\},\{\emptyset,\{\Diamond\}\}\} ;

with \sharp\mathcal{P}(\mathcal{P}(X))=4. In fact, one can iterate this n times and find \sharp\mathcal{P}^{(n)}(X) = 2^n. It is interesting to note that the empty set played a role in increasing the cardinality of the nested power sets. We will later show that one can do better than this to build up what we know as numbers.

To end this part, I add a cautionary note that not all collections of objects can form a set. Consider the Russell set:

R=\{x:x\notin x\} .

Let’s ask: does R satisfies the property given? If yes, then R\notin R but then by the set definition R\in R – giving a contradiction. Suppose the answer is no then, which means R\in R. However the set definition implies R\notin R – again, a contradiction. This is essentially known as the Russell’s paradox. It’s resolution? Perhaps, it is too much to impose R to be a set, but R is said to be a class (will not elaborate here). Essentially the idea is to differentiate the use of a symbol and the meaning of the symbol. Alternatively, one could axiomatize set theory using Zermelo-Fraenkel axioms, which is beyond the scope of the lecture.

References

I list here some of my reading materials without attempting to show where I have used them. Go read them, if you have interest.