Questions tagged [general-topology]
Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.
60,025 questions
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Topology of the End space of a Metric Space [closed]
Bridson-Haefliger, Page 144. The topology of the end space is described here through convergence of sequences. My question is, how do I recover the open sets (at least a basis of the topology) from ...
- 31
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Attribution for Metrization Theorem
Let $f:X \rightarrow Y$ be a continuous surjection with $X$ a compact metric space and $Y$ a Hausdorff space. Then $Y$ is metrizable.
Does this theorem have a name? Who first proved it? Just ...
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Doubt about effect of Standard Scaler on persistent homology
I’m currently studying persistent homology, and I’m quite new to the topic — so please forgive me if this is a naive question.
I’m trying to analyze $2D$ data where each axis represents quantities ...
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Orbit space is homeomorphic to $[0,\infty)$ using the hint and uniqueness of quotient space
Problem 3-24 from Topological manifold by John Lee. I am aware that this question was asked somewhere but there was no selected answer. AND non of the attempts uses the hint and material from the ...
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Why do we impose the Euclidean topology in the definition of a manifold? [duplicate]
In the definition of a (topological) manifold, we implicitly assume that the Euclidean topology is defined on $\mathbb{R}^n$. But could we not work with other topologies on the real numbers to define ...
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Constructing Alexander's Horned Sphere as an inverse limit space
I am studying dynamical systems and wild spaces, and I'm interested in the title question to realise the Horned Sphere as an attractor for a homeomorphism.
I am aware of a result by Morton Brown on ...
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Is the projection of a Borel subset of $\mathbb{R}^2$ always a Borel subset of $\mathbb{R}$? [duplicate]
Problem.
Let $A$ be a Borel subset of the plane, $A \in \mathcal{B}(\mathbb{R}^2)$.
Denote by $\pi^1A$ the image of $A$ under the projection onto the $x$-axis:
$$
\pi^1A = \{x \in \mathbb{R} : \exists ...
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Is this a homeomorphism between two sets?
I’m trying to check if the following function defines a homeomorphism between two sets.
Let
$$
X_1=\{(x,y)\in\mathbb{R}^2\mid 0\le y\le 1-|x|,\; y\ne 1\}
$$
and
$$
X_2=\{(x,y)\in\mathbb{R}^2\mid x^2+y^...
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If every subsequence of a net into a first-countable space converges, does the net itself converge? [duplicate]
Let $D$ be any directed set, let $X$ be any first-countable topological space, and let $\{x_\alpha\}_{\alpha \in D}$ be a net taking values in $X.$ Suppose that every subsequence of the net converges. ...
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Relation between Homotopy extension property and exact sequences
I am starting to study homotopy theory and there's a relation between the Homotopy extension property (HEP) and exact sequences that I'm not understanding.
For context let me first give the ...
- 525
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Cancellation property for self-products of spaces [closed]
The cancellation property (if A x C = B x C, then A = B) has been studied in a variant of settings, for example the category of spaces, groups, rings, etc. I am interested in a variant cancellation ...
- 1,691
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Basis of a topology - Equivalent definitions confusion
I know that similar questions were already asked here, here and here. Unfortunately I wasn't able to understand the equivalence of definitions from any of answers (to any of the linked questions). I ...
- 113
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Prove that the Zariski topology and the co-finite topologies coincide as topologies over $\mathbb{F}$
Let $\mathbb{F}$ be a algebraically close field with $\mathrm{char} \left( \mathbb{F} \right) =0$ and denote $\tau_{z} , \tau_{f}$ the Zariski and co-finite topologies on $\mathbb{F}$ respectively. I ...
- 113
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Finite and non-compact (non-Hausdorff) cell complex?
$\DeclareMathOperator{\int}{int}$Let $(X,(e_\alpha)_{\alpha \in I})$ be a finite cell complex, i.e. $I$ finite. If $X$ is Hausdorff then the closures of the cells $\bar e_\alpha$ are compact hence $X$ ...
- 7,066
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Characterizing Sets that Allow Paths to be Drawn in Mixed Rational/Irrational Dust
So I accidentally posted this problem on MO that turned out to be standard:
https://mathoverflow.net/questions/501448/example-of-connected-locally-connected-metric-space-that-isnt-path-connected
In ...
- 734
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A nonempty convex set in R^k is connected. [duplicate]
I'm following Walter Rudin's "Principles of Mathematical Analysis". This was one of the exercises of Chapter 2. I would like some feedback on the proof and know if it is correct.
Proof:
Let $...
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Interplay between metric and measure structure
On the real line the absolute value plays a dual role: a metric (by letting $d(x,y)=|x-y|$ and the seed of a measure (by letting the measure of an interval between $a$ and $b$ be the distance between $...
- 7,469
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Sufficient Properties for Metric Spaces so that connected+locally connected implies path-connected [closed]
Let's call a space clc if it's metrizable, connected and locally connected. By the MMM Theorem, if $X$ is complete then $X$ is path-connected (and locally path-connected).
Are there any known ...
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443
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Fundamental group of spaces of diagonalizable matrices
I have been studying matrix topology and some properties of subsets of $\mathfrak{M_n(\mathbb{K})}$ where $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}, n \in \mathbb{N}$ that we don't mention as much as ...
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Do the boundary points of a subset of a Euclidean space hold a special meaning in its relative topology?
So when considering the subspace topology, the subset becomes a clopen set, and all the boundary points it contains become inner points. However I am wondering if there are any topological properties ...
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How to prove that component group is totally disconnected
Let $G$ be a topological group, and $G^0$ be its identity component (i.e. $G^0$ is the connected component of $G$ containing $e$). Let $\pi_0(G):=G/G^0$, which is called a component group. It is said ...
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Is there a name for this convergent net-like, filter-like, or filterbase-like object?
This question is similar to, but ultimately distinct from, a previous question of mine. I am familiar with the sequences, filters, filterbases, and nets that can converge about a point — these terms ...
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Inner automorphism of finite dimensional algebra
There is a claim on Baker's Matrix Groups about inner automorphisms which states the following:
Proposition 4.49: Let $A$ be a finite dimensional (normed) algebra
over $\mathbb{R}$. Then the inner ...
- 353
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270
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continuous hex theorem: Path connecting opposite sides in square covered by two closed sets
If the square $[0,1]^2$ is covered by two closed sets $A\subseteq [0,1]^2$, $B\subseteq [0,1]^2$ with $[0,1]^2 = A \cup B$, is there always a path inside $A$ from lower edge $[0,1]\times \{0\}$ to the ...
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When is it possible to continuously vary two parameters $s$ and $t$ such that $f(s)=g(t)$ always holds
While studying a geometry problem I came across this interesting question.
First of all, let $I=[0,1]$. Now, let $f,g:I\to I$ be continuous functions such that $f(0)=g(0)=0$ and $f(1)=g(1)=1$ (they ...
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