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In a homework set I am assigned there are three structures we need to prove are not manifolds; the space of square matrices with $0$ determinant, the space of quintic polynomials with three real solutions, and two intersecting lines in the plane.

Let's look at the last one. It's well known. The argument is outlined in Tu's book: if $U$ is any neighborhood of the point $p$ at which the lines intersect then the property that a manifold is locally euclidean of dimension $n$ leads to a contradiction, because the number of connected components is not preserved.

But a classmate brought up a good question that stumped the graduate TA as well as me; when proving a space is not a manifold, are we proving that a topological space is not a manifold? Or that some underlying set is not a manifold when given any atlas, or any topological structure? After all, it was noted that $\mathbb{R}$ with the indiscrete topology $\{\mathbb{R}, \varnothing\}$ is not a manifold.

So how do we know that any open set containing the point of intersection for the cross necessarily contains points near that point, and thus the lines? This assumes that the open set has the form of an open set in the topology inherited from $\mathbb{R}^2$, but this is not made explicitly clear anywhere.

Sammy Black
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doorkicker
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    From a set-theoretical point of view, all these examples are in bijection with $\mathbb{R}$, so they can be endowed with some manifold structure. The problem probably assumes at least that the manifold structure must induce the same topology as the Euclidean one. – Aphelli Oct 04 '22 at 00:26

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You don't have to talk about atlases at all to define a topological manifold: a manifold is a topological space satisfying some topological conditions, namely 1) Hausdorff, 2) locally Euclidean, 3) second-countable (or paracompact or whatever your favorite version of this definition is). So you prove that a topological space is not a manifold by proving that one of these conditions fails for a given topology (probably locally Euclidean, in interesting examples).

In these examples the topology is implicitly being taken to be the subspace topology with respect to the Euclidean topology. If you don't make this assumption then the problems are trivial.

Qiaochu Yuan
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