The motivation for this exposition is threefold:
- It helps illustrate the mantra that intuition is as equally important as algebraic manipulation.
- It relates to a fundamental concept in mathematics, namely morphisms (from category theory).
- It helps explain why compact spaces and Hausdorff spaces are important.
The following facts are readily found in textbooks; the tenet of this exposition is that it is impossible, without further thought, to appreciate any of these facts.
A topological space is compact if every open cover has a finite subcover.
A topological space is Hausdorff if every two points can be separated by neighbourhoods.
A map (i.e., a continuous function) between topological spaces and is a homeomorphism if it is bijective and its inverse is continuous.
If is compact and is Hausdorff then a bijective map is automatically a homeomorphism.
- How can a bijective map not be a homeomorphism?
- How can compactness possibly have any relevance to whether a map is a homeomorphism?
- Why the Hausdorff assumption on ?
- How does this relate to “morphisms”, as mentioned at the start of this exposition?
Before addressing these questions, a standard proof of the above result is given, the purpose being to emphasise that the proof, on its own, does not directly answer any of these questions. Memorising the proof is essentially worthless. (Important are being able to derive the proof from scratch and being able to understand why the result should indeed be true.)
Showing is continuous means showing the preimage of every open set is open, or equivalently, showing the preimage of every closed set is closed. Since is bijective, the preimage under of a set is simply . Hence it suffices to prove that is closed (the image of every closed set is closed). Let be closed. Since is compact, must be compact. The image of a compact set under a continuous function is itself compact, that is, is compact. A compact subset of a Hausdorff space must be closed, that is, is a closed subset of , proving is closed and therefore that is continuous, as required.
While basing topology on open sets works very well from the perspective of being able to give concise proofs such as the one above, it is perhaps the worst way of introducing topology. It is noteworthy that when topology was being developed, it took decades for the importance of “open sets” to be recognised.
Let’s start from scratch and try to understand things intuitively. One approach (but certainly not the only approach) to understanding a topological space is understanding what converges to what. In special cases, such as when the topology comes from a metric, it suffices to consider sequences (the curious reader may wish to read about sequential spaces). In general, nets must be considered instead of sequences, nevertheless, for the purposes of answering the questions above, it suffices to restrict attention to sequences. (It is often enough to understand intuitively why a result is true in special cases, leaving the proof to justify the result in full generality.)
Why a Bijective Map need not be a Homeomorphism
By restricting attention to sequences (and therefore to nice topological spaces), the definition of continuity becomes: is continuous if, whenever , it holds that . Therefore, a bijective map is a homeomorphism if and only if implies .
Whereas bijectivity just tests whether distinct points are mapped to distinct points, being a homeomorphism means nearby points must be mapped to nearby points, in both directions, from to and from to . (That is, a homeomorphism preserves the topology.)
This suggests the following example. Let be the interval and let be the unit circle . Let be the function sending to . Then is continuous because it sends nearby points in to nearby points in . But is not continuous because the point is sent to the point but a point just below , say for just less than , is sent to a point far away from . (The reader should pause to understand this fully; drawing a picture may help.)
The above example is quintessential; an injective map is a homeomorphism only if it is not possible to map a “line ” to a “loop “. In detail, let be a sequence of points that either does not have a limit point, or that converges to a point that is different from . Such a sequence can be obtained by considering the location at regularly spaced intervals of an ant walking along a path in that does not eventually return to where it started. It is therefore referred to loosely as a “line”. Let it loosely be said that this “line” gets mapped to a “loop” if . If it is possible for a line to be mapped to a loop, then cannot be a homeomorphism. (Draw some pictures!)
To summarise, injectivity of means that if then . For to be a homeomorphism requires more: if then it must be that . Whereas injectivity just means the preimages of distinct points are distinct, a homeomorphism means the preimages of convergent sequences with distinct limits are convergent sequences with distinct limits. Perhaps the best way of visualising this though is asking whether a “line” can be mapped to a “loop”.
Relevance of the Domain being Compact and the Codomain being Hausdorff
Let be a “line”, as defined above. There are two cases to consider: either does not have a limit point, in which case it will be seen that compactness of is relevant, or has a limit, in which case it will be seen that being Hausdorff is significant.
When dealing with sequences, the role of compactness is guaranteeing every sequence has a limit point (or equivalently, that a convergent subsequence exists). Therefore, if is compact, it is not possible for to lack a limit point, and this option is immediately off the table.
If it is possible for a “line” to be mapped to a “loop” despite being compact then it is possible to achieve this assuming has a limit (by considering a convergent subsequence if necessary). Henceforth, assume yet . Continuity of implies . When dealing with sequences, the role of the Hausdorff requirement is to ensure limits are unique. (Prove this for yourself, or look up a proof that limits are unique in Hausdorff spaces.) Therefore, being Hausdorff means since converges to both and . Since is injective, this means , contradicting being a “line”. Therefore, it is impossible for a “line” to be mapped to a “loop”; every injective map from a compact space to a Hausdorff space is automatically a homeomorphism.
The above has hopefully taken away a bit of the mystery of why compactness and the Hausdorff condition enter into the picture.
- An injective map (on a nice topological space where it suffices to reason with sequences) is a homeomorphism if and only if it is not possible for a “line ” to be mapped to a “loop “. (A line and a loop are topologically distinct whereas the purpose of a homeomorphism is to preserve topological structure.)
- If is a loop in , that is, , then consider its preimage . Unless there is some reason for to converge, the fact that is injective is powerless to prevent from being a “line”, i.e., it may well be the case that is not a homeomorphism.
- If is compact then has a limit point. By considering a subsequence if necessary, assume without loss of generality that . Continuity of implies .
- If is Hausdorff then limits are unique, therefore, . Therefore, the preimage of a loop is a loop, not a line. The topology has been preserved, hence is a homeomorphism.
- The above is “intuitively” why the result holds, yet the formal proof based on open sets is shorter, more general but devoid of intuition (at least for beginners); this is the joy of mathematics.
Very briefly, it is remarked that structure-preserving functions are called morphisms (in category theory). Two sets are equivalent if there exists a bijective function between them. As more structure is given to these sets, requirements stronger than just bijectivity are necessary if the structure is to be preserved.
In linear algebra, two vector spaces are the same if there exists a bijective function between them that is linear and whose inverse is linear; the linearity of and is what preserves the structure. Interestingly, if is linear and bijective then its inverse is automatically linear.
In topology, a continuous function preserves structure in one direction, therefore, to preserve structure in both directions requires a bijective that is continuous and whose inverse is also continuous; as introduced earlier, such a function is called a homeomorphism. Unlike the linear case, being bijective and continuous is not enough to imply is continuous.