4k views

### Is every manifold a metric space?

I'm trying to learn some topology as a hobby, and my understanding is that all manifolds are examples of topological spaces. Similarly, all metric spaces are also examples of topological spaces. I ...
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### Topological manifolds (dimension)

I am taking an introductory course to topology and the professor defined a topological manifold of dimension $n$ if it is hausdorff and if for every point $x$ there exists an open set $U$ around $x$ ...
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### Can a topological manifold be non-connected and each component with different dimension?

These are two definitions in page 48 of the book an introduction to manifolds by Loring Tu. Definition 5.1. A topological space $M$ is locally Euclidean of dimension $n$ if every point $p$ in $M$ has ...
833 views

### Embedding, local diffeomorphism, and local immersion theorem.

Suppose $f: M \to N$ is smooth and an immersion, i.e $df_p : T_p(M) \to T_p(N)$ is one-to-one. Since $f$ is an immersion, we have the following theorem, $\textbf{Local Immersion Theorem:}$ Suppose ...
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### What is/are the definitions of local diffeomorphism onto image?

In summary: Actually, I think the confusion arises from a distinction between (local diffeomorphism)-onto image and local-(diffeomorphism onto image). See (C1) at the end. Firstly, I believe this is ...
### Is $[0,1) \cup \{2\}$ a manifold with boundary? My issue is the $2$.
This has been asked about here: Understanding topological and manifold boundaries on the real line, and Sharkos said Personally I'd say $M$ wasn't a valid manifold with boundary because the $\{2\}$ ...