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 is a collection of subsets of called *open sets* satisfying

- and are elements in
- Closure under finite intersections

If , then . - Closure under arbitrary unions

If is any collection of elements in (labelled by index set , then .

The pair is called a *topological space*. Often the case, physicists tend to refer to 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 with an ordered relation . Intervals of form needed subsets. For example,

is an open interval.

is a closed interval.

Note that form the open sets while form the closed sets. The open intervals in fact form the standard topology on .

Closed sets are different creatures but not unrelated. A set if topological space is closed if set is open i.e. . For example, is closed since 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 be a topological space. Then

- Arbirary intersections of closed sets in are closed.
- Finite union of closed sets in are closed.

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

To prove the first point, consider of closed sets. By deMorgan’s law, we have

.

But is open and arbitrary union of open sets are open. Hence is open, which then implies is closed.

For the second point, consider closed sets for . By de Morgan’s law, we have

Since is open and finite intersections of open sets are open, then is open or is closed. QED.

A set can also be neither open nor closed. Consider with the standard topology. The intervals and are both open but is not open and it is also not closed. This can be understood in the following way; its complement being

is the union of an open set and a closed set. Thus the said property of .

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 as a topological space with the standard topology. Now is open in since is closed. But then is closed in ; 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 , 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 of . The result would give the set , which is not open (hence failing the condition).

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

is an open disk centred at and this serves as the open sets for . Taking its closure,

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 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 .

Let . Checking on its topology property e.g.

which seems to be fine but the following property

fails. Thus is not a topology on .

If we let , then a check on its topological properties shows that it is a valid topology on e.g.

Next, we can consider . It can be easily checked that forms a topology on . The topology is called a *discrete topology* on and is called a *discrete space*.

We could also form the topology and this is said to be the *trivial topology* on . 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 and we say this as is coarser than and is finer than . 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 with Euclidean metric whose open set is the open disk. We can equip the space with a different metric, the taxicab metric, whose open set will be an open square. One can inscribe the square in the disk indicating the -induced topology finer that the -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 is *Hausdorff* if given any pair of distinct points , there exists neighbourhoods (open sets cointaining respective points) of and of such that . 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.

*axiom:* If , there exists open set such that

.

A space is called a *Kolmogorov space*.

*axiom:* If , there exist open sets such that

.

A space is called a *Frechet space*.

*axiom:* If , there exist open sets such that

.

This is the case of the *Hausdorff space*.

*axiom:* If is a closed set with , there exist open sets such that

.

*axiom**:* If are disjoint closed sets in , there exist open sets such that

.

*axiom**:* If are separated sets (i.e. ) in , there exist open sets such that

.

Combining conditions gives what is known as *regular space* and combining gives a *normal space*. We also have the space where the axiom is supplemented by the condition , 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.

- James R. Munkres, “Topology“, (Prentice Hall, 2000)
- John M. Lee, “Introduction to Topological Manifolds“, (Springer, 2000)
- G.F. Simmons, “Introduction to Topology and Modern Analysis“, (McGraw Hill, 1963)
- John G. Hocking & Gail S. Young, “Topology“, (Dover, 1988)
- Lynn Arthur Steen & J. Arthur Seebach Jr, “Counterexamples in Topology“, (Dover, 1995)

Posted by Nurisya MohdShah on June 5, 2012 at 5:13 am

I just know that there exist a ‘clopen’. Thanks!