## Topological Manifolds

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.

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