Linked Questions

17
votes
4answers
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 ...
18
votes
2answers
1k views

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 $x$ ...
5
votes
3answers
530 views

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 ...
7
votes
2answers
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 ...
9
votes
1answer
319 views

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 ...
3
votes
2answers
257 views

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 $\{2\}$ ...
7
votes
1answer
213 views

Why do connected oriented manifolds have compactly supported forms with integral one but with support contained in a given open proper subset?

My book is From Calculus to Cohomology by Ib Madsen and Jørgen Tornehave. It seems to be claimed In the proof of Lemma 10.17: For every proper open subset $W$ of $\mathbb R^n$, there is an $\omega ...
1
vote
1answer
718 views

Dimension of disjoint union of manifolds

While it is clear that a disjoint union of two $d$-manifolds is a $d$-manifold, it is not clear to me if the disjoint union of a $d_1$-manifold and a $d_2$-manifold is still a manifold and if yes ...
4
votes
2answers
311 views

Confusion with immersions, embeddings, local homeomorphisms, and local diffeomorphisms.

Definitions. A local homeo/diffemorphism is a continuous/smooth map $f:X\to Y$ such that there's an open cover $(U_i)$ of $X$ for which $f|_{U_i}:U_i\to fU_i$ is a homeo/diffeomorphism. A topological/...
4
votes
2answers
236 views

Does the Riemannian metric induced by a diffeomorphism $F$ exist for a reason other than the existence of vector field pushforwards?

My book is Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), a sequel to both Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (...
2
votes
1answer
233 views

Understanding topological and manifold boundaries on the real line

Let $M$ be the subset $[0,1)$ $∪ $ {$2$} of the real line. Find its topological boundary $\mathrm{bd}(M)$ and its manifold boundary $\partial M$. I know that to find the topological boundary, I need ...
2
votes
2answers
172 views

Immersions are open maps

Let $M,N$ be manifolds with $\dim M = \dim N$. If $f:M\to N$ is an immersion then $f$ is open. I thought that I have solved it, but then I thought there could be a mistake: Let $p\in M$. As $f$ is ...
-1
votes
2answers
144 views

Do homeomorphic smooth manifolds, like diffeomorphic ones, have the same dimension? [duplicate]

For smooth manifolds $A$ and $B$ with respective dimensions $a$ and $b$. If $A$ and $B$ are diffeomorphic, then $a=b$. I guess the same is true for homeomorphic topological ($C^0$, I guess) manifolds (...
0
votes
1answer
215 views

What is the induced orientation on a 1-manifold with boundary that is the image of closed interval under a smooth immersion?

My book is An Introduction to Manifolds by Loring W. Tu. Pictured below is the last example from Section 22, Manifolds with Boundary. I have been trying to wrap my head around this for about 2 hours (...
0
votes
1answer
70 views

Can manifold subsets always be made into submanifolds?

Related question: Are manifold subsets submanifolds? Assume all manifolds, topological or smooth discussed here have dimensions and do not have boundary. Let $A'$ and $B'$ be sets with $A' \subseteq ...

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