# Abstract Nonsense

## Topological Manifolds (pt. I)

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.

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Motivation

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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)!

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While this level of generality is nice, because it allows us to consider vast amount of spaces which would be closed off to us if we were more restrictive (for example, $\text{Spec}(\mathbb{Z})$ isn’t even $T_1$!). That said, trying to prove theorems in this general of a setting is damn hard. In fact, saying anything meaningful about general topological spaces is a herculean effort! It is precisely because of this that, once we have the notion of topological spaces, we start to impose conditions on the spaces we actually want to consider. This is why you rarely see a theorem of the form “Let $X$ be a topological space, then…” instead you see something to the effect “Let $X$ be a locally connected, paracompact, $T_4$ space, then…”.

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While we can often deal with some serious non-compactness, and some slightly connectedness pathology in a space, we often times have the most difficulty with a space failing to satisfy the correct separation axioms. For example, it is pretty universal amongst all topologist to want to assume that your space is at least Hausdorff. That said, what would make us happy is if satisfied the whole kitten caboodle–that our space be as nice (separation wise) as possible. This often comes in the form of restricting our focus to metric spaces, or more accurately to metrizable spaces.

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Now, while metrizable spaces are extremely nice in some ways (they are perfectly normal Hausdorff, for example) and this makes us happy, they still have the nasty habit of surprising us with unexpected pathologies. For example, we expect our spaces to be locally connected, or locally compact. A very naive thing to think is that locally around a point should look “large”, uncountable. Of course, it is easy to break all of these using just metric spaces. It is not too hard (I’ll leave it to you construct such a space!).

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All of the desired qualities described above have a common denominator–we want our space to locally look “nice”. Well, what does nice mean? Well, you could parrot back the things we’ve already said: connected, compact, “large”, etc. That said, why are these the properties that our intuition tells us topological spaces locally should have? It is all because we are using a particular (very small) class of spaces as a measuring stick for the well-behavedness of our spaces. In particular, all of these desired local attributes of our space come from looking back to our home–the Euclidean spaces ($\mathbb{R}^n$ for some $n\in\mathbb{N}\cup\{0\}$). Namely, we want our spaces to have local properties that the Euclidean spaces enjoy. Well, continuing in our theme of trying to find extremely nice spaces, why don’t we go the whole way? Instead of saying that we only want to deal with spaces that locally have the properties of Euclidean space, why don’t we instead say that we want our spaces to locally look like Euclidean space!

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This process of filtration, of restricting ourselves to nicer and nicer spaces–spaces to have nice separation axioms, to spaces that are metrizable, to spaces that locally enjoy the properties of Euclidean space, to spaces that locally look Euclidean–is precisely the process by which one ends up with the notion of a topological manifold. How one actually defines a topological manifold is slightly more sophisticated (in what way do they locally look Euclidean?) but the basic idea is just that–topological manifolds are extremely nice spaces because they locally look like our favorite space $\mathbb{R}^n$–for some $n$.

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Topological Manifolds

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Let $M$ be a topological space. An ($n$-dimensional) chart for $M$ consists of an ordered pair $(U,\varphi)$ where $U\subseteq M$ is open and $\varphi:U\to\varphi(U)\subseteq\mathbb{R}^n$ is a homeomorphism such that $\varphi(U)\subseteq\mathbb{R}^n$ is open. Intuitively, we can think about this map as singling out an open subset of $M$ which looks like (i.e. is homeomorphic to) a crumpled, twisted up piece of $\mathbb{R}^n$ (this is $U$) and uncrumpling, untwisting it into its natural, pristine form (this is $\varphi$). If $p\in M$ and $(U,\varphi)$ is a chart such that $p\in U$ we say that $(U,\varphi)$ is a chart at $p$

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For example, if $M=\mathbb{R}^n$ and we have that $U\subseteq\mathbb{R}^n$ is some open set then the inclusion map $U\hookrightarrow\mathbb{R}^n$ is a chart.

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Consider the circle $M=\mathbb{S}^1$, and consider $U=\mathbb{S}^1-\{(0,1)\}$. Then, the map $\varphi:U\to\mathbb{R}$ defined by $\varphi(x,y)=\frac{x}{1-y}$ is a chart. Indeed, we claim that $\varphi(U)=\mathbb{R}$, certainly an open subset of $\mathbb{R}$, and that $\varphi$ is a homeomorphism. To see that $\varphi(U)=\mathbb{R}$ we merely note that $y=1-x^2$ and so $\varphi(x,y)=\frac{1}{x}$ and since every value of $\mathbb{R}$ occurs as a first coordinate in an element of $U$ we see that $\varphi$ is surjective. It’s patently obvious that $\varphi$ is continuous, it’s and it’s inverse is continuous from where it follows that $\varphi$ is a homemorphism and thus a chart. Intuitively we can think of $\varphi$ as taking the unit circle minus its crown and unfolding, stretching it until we have the entire real line.

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Let $M$ be a topological space. An ($n$-dimensional continuous) atlas for $M$ consists of a set of ($n$-dimensional) charts $\left\{(U_\alpha,\varphi_\alpha)\right\}_{\alpha\in\mathcal{A}}$ such that $\bigcup_\alpha U_\alpha=M$. A Hausdorff second countable (i.e. having a countable basis) topological space $M$ which admits a continuous $n$-dimensional atlas is called an $n$-dimensional topological manifold. The additional requirements that our space be Hausdorff and second countable is just purely out of technical necessity (in particular, the existence of partitions of unity). Really, intuitively, we should just think of topological manifolds as being spaces made up by homeomorphically gluing together subsets of Euclidean in a very strict way.

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Note that since second countability and Hausdorffness are hereditary properties for metric spaces, and $\mathbb{R}^n$ is a second countable Hausdorff metric space (of course, all metric spaces are Hausdorff!) we see that necessarily any subspace of $\mathbb{R}^n$ necessarily satisfies the Hausdorff and second countability axioms.

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It’s fairly obvious that the statement “is a topological manifold” is topological (i.e. is invariant homeomorphism). Indeed, if $f:M\to N$ is a homeomorphism of spaces and $N$ is a topological manifold with chart $\{(U_\alpha,\varphi_\alpha)\}$ then it is easy to verify that $\{(f^{-1}(U_\alpha),\varphi_\alpha\circ f\}$ is an atlas for $M$.

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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.