## Topological Manifolds (Pt. III)

**Point of Post: **This is a continuation of this post.

**Some Topological Properties of Topological Manifolds**

What we’d now like to do is discuss some of the nice topological properties that topological manifolds have. The first of the them is the fact that they have a countable cover to topological Euclidean balls (i.e. spaces homeomorphic to a ball for some ). The basic idea is simple, at each point we know that there is a neighborhood which is homeomomorphic to an open subset of . Now, this open subset of Euclidean space contains a Euclidean ball around the image of –the preimage of this ball will be a topological Euclidean ball containing . Now, we have an open cover of our manifold by topological Euclidean balls–the argument is then over once we recall that every second countable space is Lindelof and so we may find a countable subcover. Formally:

**Theorem: ***Let be a topological manifold. Then, admits a countable atlas for topological Euclidean balls.*

This is a good time to point out a non-example of a topological manifold. Now, while you may be trying to think of some crazy, pathological beast of a space, there is in fact a very nice non-manifold living (embedded) in your computer screen. Indeed, consider the topological space represented by the plus sign: . More rigorously, we can consider the infinite analogue gotten by considering the union of the two lines and sitting inside as a subspace. Why is not a topological manifold? Well, suppose for a second that it was, then by the above theorem there exists a neighborhood of the point where the two lines meet (the origin in the more rigorous version) which is homeomorphic to for some (clearly nonzero) . That said, if then is homeomorphic to minus a point. That said, has four connected components whereas minus a point has at most two connected components (two in dimension and zero in higher dimensions)–this is a contradiction.

With this theorem we can prove many, very interesting, facts about the topology of a topological manifold. For example, we immediately get the following theorems:

**Theorem: ***Every topological manifold is locally simply connected. *

**Theorem: ***Every topological manifold is locally compact. In fact, it has a countable basis of precompact Euclidean topological balls.*

Now, perhaps a more interesting (because it is not as obvious) fact is the following:

**Theorem: ***Let be a topological manifold. Then, is connected if and only if it’s path connected.*

This follows immediately from the following lemma:

**Lemma:** *Let be a locally path connected connected space, then is path connected.*

**Proof:** Let and let be the set of all points in for which there is a path from to that point. Since () it suffices to prove that it is both open and closed (and the result will then follow from ‘s connectedness). So, let . By assumption there exists some neighborhood of it such that is path connected, we claim that . To see this, let . Concatenating the paths from to , and from to we can then construct a path from to and so . It follows that is open.

Now, to show that is closed we show it is invariant under closure, and since it suffices to show the reverse inclusion. So, let . Then, by ‘s local connectedness there exists a neighborhood of which is path connected. But, since there must be some point . So, using the exact same technique described in theorem (*) again we may connect the path from to and the path from to to get a path from to . It follows that .

Thus, is open and closed and since it’s non-empty it must be that . But, that means that every point of may be connected to by a path. This clearly implies the theorem.

Last but not least we prove the, very interesting, fact that the fundamental group of a topological manifold is necessarily countable. This, once again, is due to the fact that our manifold has a countable cover by topological Euclidean balls. Note, since connectedness is equivalent to path connectedness, the independence of base point for fundamental groups comes to connected manifolds. Our proof heavily follows [1].

**Theorem: ***Let be a connected topological manifold, then is countable.*

**Proof: **We know that has a cover ty topological Euclidean balls . Let . Since our space is second countable we know that has countably many path components. For each let be a collection of representatives from each path component of . Let then, –evidently is countable. If and then let to be a path in . Fix some , we will show that is countable. Let be the set of all finite products of paths of the form as mentioned above. Call a loop at *special* if .

Evidently the set of special loops, being a subset of (which is clearly countable) is countable. If we can show that every element of is homotopic to a special loop, then we’ll be done. To this end, let be a loop anchored at . Since is compact, there exists a finite cover of the set . Thus, there exists finitely many points such that for some . Let be the map one obtains by restricting to and reparamaterizing so that it is defined on and let be such that . By definition, for each we have that and that there exists some point in the same component of –we can take . Let be a path where we take to be the constant loop at . Note then that

Now, each is a path and thus (since is simply connected) is homotopic to . Thus, we see that is a special loop. Since was arbitrary the conclusion follows.

**References:**

[1] Lee, John M. *Introduction to Smooth Manifolds*. New York: Springer, 2003. Print.

[2] Lee, John M. *Introduction to Topological Manifolds*. New York: Springer, 2000. Print.

[3] Milnor, John Willard, and David W. . Weaver. *Topology from the Differentiable Viewpoint*. Charlottesville (Va.): University of Virginia, 1969. Print.

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