# Plausibility of a Discrete Point Manifold

## Main Question or Discussion Point

Consider a discrete set of $k$ points.

First, is it a manifold? I know that a manifold is a topological space that contains a neighborhood homeomorphic to Euclidean space for each point. Can we just consider each point's neighborhood to be a set containing only that point?

Second, would the structure be orientable for $k>2$?

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Can we just consider each point's neighborhood to be a set containing only that point?
You can, but then where is the homeomorphism to an open subset of $\mathbb{R}^n$? All singleton sets in $\mathbb{R}^n$ are closed.

WannabeNewton
Every countable discrete space is a topological 0-manifold; the converse holds as well. Here a discrete space is a set equipped with the discrete topology. This is a trivial exercise so try to show it yourself. Define a topological n-manifold as a second countable Hausdorff space that is locally Euclidean of dimension n (i.e. every point has a neighborhood that is homeomorphic to an open subset of $\mathbb{R}^{n}$).

Every countable discrete space is a topological 0-manifold; the converse holds as well. Here a discrete space is a set equipped with the discrete topology. This is a trivial exercise so try to show it yourself. Define a topological n-manifold as a second countable Hausdorff space that is locally Euclidean of dimension n (i.e. every point has a neighborhood that is homeomorphic to an open subset of $\mathbb{R}^{n}$).
And to add to this: yes, the manifold will be orientable.

Every countable discrete space is a topological 0-manifold; the converse holds as well. Here a discrete space is a set equipped with the discrete topology. This is a trivial exercise so try to show it yourself. Define a topological n-manifold as a second countable Hausdorff space that is locally Euclidean of dimension n (i.e. every point has a neighborhood that is homeomorphic to an open subset of $\mathbb{R}^{n}$).
I'm not very confident with this material, so...here I go:

Lemma #1: If the space is a countable discrete space, then it is also a second countable space.
//Proof: If the space is countable and discrete, then there are a countable number of singletons. The singletons can, therefore, form a countable basis for the discrete space.$\halmos$​

Lemma #2: If the space is a countable discrete space, then it is a Hausdorff space.
//Proof: If the space is a discrete space, then the points are necessarily disjoint. $\halmos$​

Lemma #3: Each point in a discrete space is homeomorphic to $\mathbb{R}^0$.
//Proof: Homeomorphism is the identity map. $\halmos$?​

Considering lemmas 1, 2, and 3, we have established that a countable discrete space is a second countable Hausdorff space of dimension 0 with each neighborhood homeomorphic to Euclidean space of dimension 0, and therefore a 0-manifold. $\blackhalmos$

I'm not sure about the third lemma. It feels too...blunt.

@Micromass: How is it orientable? Could you, perhaps...nudge me in the right direction?
(Also...you've not responded to my personal message about operators. It's been 3 weeks. )

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WannabeNewton