All Questions
Tagged with topology or general-topology
60,043 questions
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How many vertices does $(\Bbb{T}^2)^{\#g}$ have?
Here is a description in a topology lecture note that I need help.
Since $(\Bbb{T}^2)^{\#g}$ can be represented by a $4g$-gon with edges identified according to the word $a_1b_1a_1^{-1}b_1^{-1}\cdots ...
- 17.4k
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Why don’t textbooks mention the 2-dimensional metric completion model of $S^2$?
Most differential geometry and topology books introduce the 2-sphere as the surface
$$
S^2 = \{ x \in \mathbb{R}^3 : \|x\| = 1 \}.
$$
This is fine, but it often leaves the impression that the sphere “...
- 11
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71
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Order in an open set
I need help to prove a theorem.
In the following definition $\operatorname{fr}_X(A)$ means the boundary of $A$ in $X$
Definition. Let $(X, \tau_X)$ be a topological space, let $p \in X$, and let $\...
- 151
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will EVT work in non-Hausdorff spaces where compact sets are not necessarily closed?
Suppose $X$ is a compact topological space and $f : X \rightarrow \mathbb{R}$ a function.
Suppose $X$ is not in an Hausdorff space i.e. that the Heine-Borel theorem does not work and where
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The natural map $j_{!} \hookrightarrow j_{*}$ on sheaves
Suppose $ j :U \to X $ is an open immersion of topological spaces. We know that $ j_{!}F $ is a subsheaf of $ j_{*}F$ for any sheaf $ F$ on $U$. If $ F = j^{*}G $ for a sheaf $ G $ on $ X$, then I ...
- 801
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1
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227
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Does "the preimage of a closed interval is a finite union of closed intervals" imply $f:\mathbb{R}\to\mathbb{R}$ is continuous?
Suppose I have a function $f:\mathbb{R}\to\mathbb{R}$ with the property that for any closed interval, its preimage is a finite union of closed intervals. Can I conclude that $f$ is continuous, or do ...
- 7,992
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Topology on the space of asymptotic growth rates
Let $\mathcal{F}$ be the space of all increasing continuous functions $f:\mathbb{R}^{+}\to \mathbb{R}^{+}$ s.t. $\lim\limits_{x\to\infty}f(x)=\infty$.
Consider the equivalence relation
$$
f \sim g\iff ...
- 674
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60
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Relative compact set in metric space
Below is an equivalent way of stating relatively compactness ($\bar{A}$ is compact) in a complete metric space.
$\forall \epsilon>0$, there exists a compact set $K$ such that $\forall a \in A$, $d(...
- 1,301
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68
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How to define a directed path in any space?
I'm trying to come up with a definition of a directed path $f: I \to X$ in an arbitrary space $(X, \mathcal{T})$ which has these properties:
A directed path is a path, i.e. $f$ is continuous from the ...
- 2,943
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64
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$\mathbb{R}P^1$ an unclear homeomorphism [closed]
$\mathbb{R}P^1$ should be homemorphic to $S^1$ via $f:z\mapsto z^2$. But I do not follow how an element in $\mathbb{R}P^1$, denote it by $[x:y]$, is calculated to be $z^2$. I think, though I do not ...
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79
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$\emptyset$ is not connected. Is this really true? (Takeshi Saito's article in Sūgaku Seminar.) [closed]
I am currently reading the article by Takeshi Saito published in the November 2025 issue of Sūgaku Seminar.
He wrote $\emptyset$ is not connected.
Is this really true?
I don't know Category Theory at ...
- 10.4k
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71
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Using the dominated convergence theorem with nets
I have doubts as to how the dominated convergence theorem is applied in contexts where you have nets and not sequences. For context, my question is related to an answer to this question https://math....
- 3,355
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81
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Is it ok to identify $L\subset\mathbb{R}\times\mathbb{R}$ with $\mathbb{R}$ and describe the topology on $L$ in terms of the topology on $\mathbb{R}$?
I am reading "Topology Second Edition" by James R. Munkres.
Munkres does not define homeomorphisms between topological spaces in the pages leading up to the following Exercise 8. What kind ...
- 10.4k
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455
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A subset of a rectangle that 'blocks' every curve that goes from right to left must connect upper and lower sides
I'm trying to find a proof for the following assetion: Given a rectangular region $R$ and a subset $A$ of $R$, if every curve that starts at the left side of $R$ and ends at the right side intersects $...
- 333
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Let be X a metric space, A and B closed sets such that the union and intersection are arcwise connected, show that A and B are arcwise connected. [closed]
I have seen the same question, but talking about connected spaces. I would like to know how to do this with arc wise connected spaces.
