Point of Post: In this post we define the notion of a topological manifolds, discuss some of their importance, and prove some elementary topological facts about them.
Anyone who has taken a general topology course before can tell you the following simple fact: topological spaces can get messy. It is not hard to create topological spaces which completely break our intuition for what a topological space should be! It’s definitively possible that a topological space doesn’t locally look like connected space (i.e. isn’t locally connected), doesn’t locally look like a compact space (i.e. isn’t locally compact), and (in a more dramatic way) it doesn’t even have to be able to distinguish points (i.e. doesn’t have to be Kolomogorov)!
In this post we will discuss some of the local properties manifolds attain from dimensional Euclidean space. It should be intuitive that all of the local attributes should transfer since manifolds “locally look like Euclidean space”. We formalize them in what follows.
We start with a theorem which when put together with the definition of manifolds will give us an amazing revelation.
Theorem: Let be a -manifold. Then, is locally compact.
Proof: Let be arbitrary. As was proven earlier every manifold has an open base of Euclidean balls, and so let be the guaranteed Euclidean ball containing . So, by assumption for some open ball . So, by the regularity of there exists some open ball such that . So, is a neighborhood of in . Also,
And so, in particular
is a homeomorphism and so is compact in . Thus, is a precompact neighborhood of in . But, since compactness in a subspace implies compactness in the ambient space and is open we see that is in fact a precompact neighborhood of in . The conclusion follows.
The following corollary, to me, was astounding albeit intuitively clear.
Corollary: Since every locally compact Hausdorff space is regular and every second countable regular space is normal we see in particular that every manifold is second countable and normal and thus metrizable by Urysohn’s embedding theorem.
Also, it follows that every manifold which is not compact has an Alexandroff compactification.
We next discuss some issues related to local connectedness/path connectedness.
Theorem: Let be a -manifold. Then, is locally path connected and consequently locally connected.
Proof: Let be arbitrary and let be the guaranteed Euclidean ball around it. By assumption for some open ball . But, as was proven earlier every open ball in a normed vector space is convex and thus path connected. Thus, is path connected and the conclusion follows.
This was just a quick post to prove some results which are true, but need to be proven at least once.
We now begin our talk on topological manifolds (to be defined shortly). From there we will begin to study some of the effects that prior subjects have on such manifolds. For example, what are necessary and sufficient conditions for a topological manifold (now just called manifolds) to be compact? Connected? Metrizable? We will also introduce some concepts previously left out for no particular reason in that it didn’t fit. For example, we will discuss the concepts of path connectedness and quotient spaces. Both are very relevant in the realm of point-set topology but the full realization of their power is when they are coupled with the idea of manifolds. So, let us begin:
Topological Manifold of Dimension (-manifold): Let be a topological space that is : locally Euclidean of dimension , second countable, and Hausdorff. Then, we call a topological manifold of dimension (an -manifold).
We first prove a small theorem and then give some concrete examples of manifolds.
Theorem: Let be an -manifold and an open subspace of . Then, is an -manifold.
Proof: Since any subspace of a Hausdorff space is Hausdorff and any subspace of a second countable space second countable it remains to show that an open subspace of a locally Euclidean space of dimension is locally Euclidean of dimension .
To do this let be the guaranteed chart at each and let be such that and . This is clearly a chart at . Since was arbitrary the conclusion follows. .
We now give some examples
Example: Let . It is clear that being the subspace of is both second countable and Hausdorff. It remains to show that it is locally Euclidean. We only outline this procedure since the full construction can be found in any textbook on geometric topology. It is called stereographic projection. We define two maps on subspace of . So, let and and define by and given by . In the three-dimensional case this can be thought of as removing the north or south pole and smoothing out what’s rest onto the plane. It turns out that both these maps are homeomorphisms. And so, let if if take the chart at to be where and . If take the cart to be where and .
It makes sense though. That although is three dimensional since cutting out a smaller portion and examining it reveals that you are really looking at a portion of a plane (a two dimensional object) but just bent slightly.
Clearly any open subspace of Euclidean space is itself a manifold. So are tori (they are -manifolds). Also, though I will not prove it so is the real projective spaces
Our next definition is a specific kind of -manifold.
-manifold with boundary: Let be a second countable Hausdorff space such that for every there exists some neighborhood and some open such that . Then is called an -manifold with boundary. We define . We define similarly.
It will take some serious work (probably well-beyond what we’ll get into in the near future) to show that the boundary and interior of a manifold with boundary are disjoint.
For the sake of notation convenience we define to be the two point discrete space . Our next theorem completely characterizes -manifolds.
Theorem: Let be a -manifold. Then, is a countable discrete space.
Proof: We claim that is discrete space. To see this let be arbitrary. By assumption there exists some neighborhood such that where . If it’s either of the first two we are done because then and which would imply that . Otherwise, and so for some . We may assume WLOG that and since is open and continuous we have that is an open subspace of and since is open it follows that is open in . The conclusion follows.
To see that it’s countable notice that is an open cover for with no proper subcover. But, by Lindeolf’s theorem it must have a countable subcover. Thus, it follows from previous comment that must be a countable subcover. The conclusion follows.
That is all for right now, there will be plenty more to come.
We wish to begin discussing the concept of manifolds (the motivation and definition will be given in a subsequent post). We have covered to a degree much more than is necessary all the concepts need to define a manifold except one. Namely, the concept of a space being “locally Euclidean”. Intuitively a locally Euclidean space of dimension is one that “looks like” on a small enough scale. Consider our planet, it is locally Euclidean of dimension for although it is embedded in to us it looks like thus giving the age-old misconception that the world is flat. It is clear (intuitively) then that the unit sphere is locally euclidean of dimension although it is not homeomorphic to any Euclidean space (since it is compact and any Euclidean space is not). Another concrete example is with the usual topology. Clearly, given any point in this space we can find a neighborhood contained in either of these components which then clearly looks like . Once again this space is not homeomorphic to since it is not connected.
Enough with the motivation, let’s move onto the formal definitions.
Locally Euclidean of Dimension : Let be a topological space such that for every there exists some neighborhood of which may embedded as an open set in . In, other words there exists some homeomorphism where is open.
Chart: If is open and is a homeomorphism where is open we call the ordered pair a chart on and if we call it a chart at .
Remark: It is clear from the above that a space is locally Euclidean if and only if there is chart at every point of .
We first show that being an open set may be replaced with either an open ball or itself.
Theorem: Let be locally Euclidean, then given any there exists some neighborhood of such that for some .
Proof: Let be arbitrary. Since is locally Euclidean there exists neighborhood of such that there exists some homeomorphism where is open. But, since is open and there exists some . It follows that if that is a homeomorphism. The conclusion follows. .
Corollary: Since every open ball in is homeomorphic to the above is still valid if is replaced with .
It turns out that the property of being locally Euclidean is inherited by open subspaces as is shown in the following:
Theorem: Let be a locally Euclidean space of dimension and an open subspace of , then is locally Euclidean of dimension .
Proof: Let be arbitrary. Then, by ‘s assumed local Euclidean property there exists some neighborhood of such that there exists some homeomorphism . Clearly then is an open neighborhood of in and is the desired homeomorphism. .
It is of course natural to ask under what conditions does a map preserve the property of being locally Euclidean. It should come as a surprise to no one that being locally Euclidean is invariant under homeomorphism. But, as with most topological properties there is a weaker condition for this to be true. But before we may give it in it’s fullest generality we need a new definition.
Local Homeomorphism: If are topological spaces and a continuous map such that for every there exists some neighborhood of such that is a homeomorphism and is open in . If so, we say that and are locally homeomorphic (symbolized by and is called a local homeomorphism.
Theorem: Let be a local homeomorphism, then is open.
Proof: Let be open and let . By assumption there exists some neighborhood of such that is open in and is a homeomorphism. Clearly then we have that is an open subspace of and thus by assumption is an open subset of . But, since is a subspace we know that for some open set in . It follows that is open in . But, this means that is a neighborhood of contained in . The conclusion follows.
Theorem: Let be a locally Euclidean space of dimension . Then, if is a surjective local homeomorphism, then is a locally Euclidean space of dimension .
Proof: Let be arbitrary. By assumption there exists some of such that is a homeomorphism and is open in . But, by assumption there also exists some neighborhood of , some open set in , and some which is a homeomorphism. Clearly then is an open subspace of and thus using previous methods we see that is a neighborhood of in . But, clearly since is open we also have that is a homeomorphism and is open (this is clear it is an open subset of the open subspace ). It follows that is a neighborhood of and is a homeomorphism with an open subset of . The conclusion follows. .
Remark: A similar idea will appear later when we discuss overlap maps. Also, the idea of local homeomorphisms is much richer than what we have shown here. Maybe one day we will explore the concept in its entirety.
The next theorem should come as a surprise to no one.
Theorem: Let be a finite number of topological spaces such that is locally Euclidean of dimension for each . Then, under the product topology is locally Euclidean of dimension .
Proof: Let be arbitrary. By assumption there exists a neighborhood of and a homeomorphism where is open. Clearly then is a neighborhood of and and open subset of . It follows from an old theorem that given by
or equivalently (if it’s hard to picture)
is a homeomorphism (since it’s the product of homeomorphisms). The conclusion follows
We now discuss particularly nice theorem regarding a base that can be imposed on . But, first a definition.
Euclidean Ball (E.B.): Let be a locally Euclidean space of dimension and let be open and such that where is an open ball. Then, we call a Euclidean ball (E.B).
Our next theorem in effect says that the Euclidean balls are sufficient to describe ‘s topology.
Theorem: Let be a locally Euclidean space of dimension then has an open base consisting entirely of E.B.s
Proof: Let , we have by assumption there exists some neighborhood of such that for some open ball , and let be the associated homeomorphism. Define then and . It is clear that each element of is an open subspace of but since is open we have that it is an open subspace of . Furthermore, it is not to hard to see that given some that is a homeomorphism. Thus, each element of is an E.B. Finally, let . It is clear that is a collection of E.B.s in it remains to show that is a base.
To see this let be arbitrary and any neighborhood of it. Letting the notation be as in the previous paragraph we have that is an open subspace of and so is an open subspace of and so there exists some such that . It follows that and . The conclusion follows.
It is not surprising that most of the “local properties” of are inherited by locally Euclidean spaces. While we will discuss most of them in subsequent posts we prove one here.
Theorem: Let be locally Euclidean of dimension . Then, is first countable.
Proof: Let be arbitrary and let be as in the last theorem. Then, just as before there exists some such that . By the Archimedean principle there exists some such that . Define
Clearly so it remains to show that it is an open base at . To do this let be any neighborhood of . Clearly then is a neighborhood of contained in . Thus, we have that is open in and so there exists some such that . Thus, appealing to the Archimedean principle again we may find some such that . Clearly then and since it’s also in the conclusion follows.
This ends our discussion for now. Next time we will talk about manifolds.