Definition of a locally Euclidean space

Question

According to Wikipedia: A topological space $(X,\tau)$ is called locally Euclidean if there is a non-negative integer $n$ such that every point $p$ in $X$ has a neighborhood $N$ which is homeomorphic to $\mathbb{R} ^n.$

But I am not sure what this means to say that $N$ is homeomorphic to $\mathbb{R} ^n$. Does it mean that the subspace topology of $(X,\tau)$ on $N \subseteq X$ is homeomorphic to $\mathbb{R} ^n$ (with Euclidean topology) ?

Answer 1

Yes, that is exactly what it means. In general, we often suppress mention of the subspace topology when referring to a set as a topological space if that set is a subset of exactly one topological space considered so far.

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Incidentally, I have a slight stylistic quibble with that definition. In my opinion the following is more natural:

$(X,\tau)$ is locally Euclidean iff for each $x\in X$ there is some $U\in\tau$ with $x\in U$ and some $V\subseteq\mathbb{R}^n$ which is open in the usual topology and $U$ with the subspace topology from $(X,\tau)$ is homeomorphic to $V$ with the subspace topology from $\mathbb{R}^n$.

It's a good exercise to show that this is equivalent to the definition given above, the point being that every open set in $\mathbb{R}^n$ contains an open set homeomorphic to $\mathbb{R}^n$. One advantage of this second definition is that it is frequently more convenient (it's a priori easier to show). But there's a more substantive point as well. Let's say we want a general notion of "$(X,\tau)$ is locally homeomorphic to $(Y,\sigma)$." In general, $(Y,\sigma)$ may not have the same "self-similarity" property of $\mathbb{R}^n$ which makes the two definitions of "locally Euclidean" equivalent. So we get two inequivalent candidates for a general notion of "locally homeomorphic" here. And to my mind, the second is nicer; note for example that it yields an equivalence relation on topological spaces.