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Results tagged with manifolds
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user 167587
A manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n.
14
votes
2
answers
863
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Must a space that is locally injective image of $\mathbb{R}^n$ be a manifold?
Suppose $X\subseteq\mathbb{R}^m$ s.t. for any $x\in X$ and any open $U\subseteq\mathbb{R}^m$ that contains $x$, there exists a smaller open set $V\subseteq U$ also containing $x$, so that $V\cap X$ is …
- 623
0
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Must a space that is locally injective image of $\mathbb{R}^n$ be a manifold?
I feel like this is true for $n=1$. Here is an outline. Suppose $X$ is Hausdorff and
$\{U\subseteq X\mid U\text{ is open and there exists a continuous bijection }f:\mathbb{R}\rightarrow U\}$
form a ba …
- 623