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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Continuity of functions in b-metric spaces
Let $X$ be a $b$-metric space and let $B(x,\varepsilon)=\{y\in X: d(x,y)<\varepsilon\}$. Then we know that $\mathcal{B}=\{B(x,\varepsilon): \varepsilon>0, x\in X\}$
does not define a topology on ...
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Cutting a Möbius strip in thirds. Why are the resulting strips interlinked?
It is well known that cutting a Möbius strip "in half" down the middle results in a band with two twists, homeomorphic to a cylinder. See this question for example.
If instead, one begins ...
- 3,715
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How to prove that the moduli space $\mathcal{M}_g$ is Hausdorff using Kuranishi families?
Let $ C $ be a compact connected Riemann surface of genus $ g > 1 $. I used the definition of a Kuranishi family of $ C $ as in [Teichmüller space via Kuranishi families] 1.
Using these Kuranishi ...
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How many components does an arc meet in a continuum
before I begin, I would like to provide some definitions and theorems.
Here $\operatorname{fr}_X(A)$ means the boundary of $A$ in $X$.
Definition. Let $(X, \tau_X)$ be a topological space, let $p \in ...
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Is it true that $X \times Y$ Hausdorff $\Rightarrow$ both $X$ and $Y$ are Hausdorff?
I've been trying to get my head across this problem. I think I found a proof, however I'm not sure if it is valid.
Let's assume $X \times Y$ is Hausdorff and prove $X$ is Hausdorff (then the same ...
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uniqueness of a map and universal property
I am trying to prove the following statement:
For any two non-empty sets $X,Y$, an equivalence relation $E$ on $X$, and an injective function $h:X/E\to Y$, show that there exists a unique function $H:...
- 636
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Lattices in universal cover of $SL(2,\mathbb{R})$
Let $\Gamma$ be a lattice in the universal cover G of $ SL(2,\mathbb R)$ and $p$ the covering map onto $ SL(2,\mathbb R)$. Is it necessarily true that $p(\Gamma)$ is again a lattice in $ SL(2,\mathbb ...
- 355
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Topological group and neighborhood base
I am reading J.S. Milne’s notes for Fields and Galois theory (v5.10). I have a question about the proof of the following proposition (page 94. Proposition 7.2).
Let $G$ be a topological group and let ...
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Terminology for local contractibility: "locally contractible" vs "strongly locally contractible" vs "semi-locally contractible"
I'm working on formalizing locally contractible spaces in Mathlib (the mathematics library
for the Lean theorem prover), and I've encountered conflicting terminology in the literature
regarding local ...
- 277
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Smallest topology on $X$ containing a collection of subsets of $X$. [closed]
The problem is to show that $\mathcal{T}_1$ is the smallest topology on $\mathbb{R}$ containing $\mathcal{B}_1$, where $\mathcal{T}_1$ and $\mathcal{B}_1$ are defined as follows:
Let $\mathcal{T}_1$ ...
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Definition of homotopy (exercise)
I am trying to do the following exercise:
Given a compact, Hausdorff topological space $X$ and a metrizable topological space $Y$ (with metric $d_Y$), let $C(X,Y)$ be the space of continuous functions ...
- 859
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Is the quotient of the subspace homeomorphic to the subspace of the quotient?
Suppose $X$ is compact Hausdorff and $\mathcal P$ is a partition of $X$. Define the map $\pi:X \to \mathcal P$ taking each point to the unique partition element containing it. Give $\mathcal P$ the ...
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Non-simply connectedness of the Moebius group [duplicate]
Assuming that a map of the form: $a\in SL(2,\mathbb{C}) \rightarrow m[a]$ with $m[a](z)=\frac{a_1z + a_2}{a_3z+a_4}$ is a group homomorphism, it is easy to show that this mapping is not bijective, ...
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Since rings of continuous maps exist why in (co)homology do you never see chain complex $C_n =$ certain ring of continuous functions "of degree $n$"?
I'm new to homological algebra. Just wondering why we never seem to see the involved chain complexes defined simply to be $C_n=$ continuous maps "of degree $n$" where the context makes the ...
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$f$ is continuous iff $f_{1},\dots,f_{n}$ continuous in Metric Spaces
I`m trying to do the following exercise for my general topology class:
Let $(X,d)$ a metric space and $B\subset\mathbb{R}^n$ with euclidean
metric. Let $f:X\to B$ and application determined by $f_{i}...
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Why is this generalized Gauss map 'clearly' continuous (in Milnor Characteristic Classes)?
In Milnor's book Characteristic Classes, Chapter 5, he defined the generalized Gauss map as follow:
$\overline{g}:M \rightarrow Gr_{n}(\mathbb{R}^{n+k})$, where $M$ is a smooth manifold in $
\mathbb{R}...
- 21
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Function spaces, uniformities, and k-spaces
I'm looking for a resource that covers the interplay between function and uniform spaces and k-spaces. All the texts I've seen so far cover one or two of the three, but never the full combination in ...
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A very dense subset of a topological space $X$ has same Krull dimension as $X$?
Definition 1. Let $X$ be a topological space. The Krull dimension $\dim X$ of $X$ is the supremum of all lengths of chains $ X_0 \supsetneq X_1 \supsetneq \cdots \supsetneq X_l$ of irreducible closed ...
- 4,056
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What are the sections of the forgetful functor $\operatorname{TopVec} \to \operatorname{Vec}$?
Let $\operatorname{Vec}$ be the category of $\mathbb{R}$-vector spaces and let $\operatorname{TopVec}$ denote the category of Hausdorff topological $\mathbb{R}$-vector spaces. There is an obvious ...
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How to embed the “Metrizable tangent disc topology” into Euclidean space?
Metrizable tangent disc topology, first appears in example #83 in Counterexamples in topology, S&S, as well as in pi-base S75, is the space $X =\,\bigl( \mathbb R \times (0, + \infty) \bigr) \cup (...
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Why does there exist no homeomorphism from $I \times I$ to the topological comb?
Hatcher claims that there exists no homeomorphism $I \times I$ to $I \times \{0\} \cup A \times I$ where $A = \{0, 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots \}$ on the basis of forward ...
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Is the underlying vector space of a condensed $\mathbb{R}$-vector space a topological $\mathbb{R}$-vector space?
According to the answer given at What exactly is a condensed $\mathbb{R}$-vector space? , a condensed $\mathbb{R}$-vector space is an internal module object in $\operatorname{CondSet}$ over the ...
- 4,822
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Showing that Smirnov compactification is a compactification
This is part of the proof that proximities on a topological space $X$ are order-isomorphic to compactifications of $X$. See wikipedia for definition of proximity space. Here I am defining proximity ...
- 15.9k
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Pushout of topological spaces, the torus, quotient topology
My professor's notes are a little bit confusing.
In particular, I don't understand the concept of "pushout".
Here is my professor's notes:
Let $X=[0,1]×[0,1]$, and let's consider the ...
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Mesurable selection theorem for product of Polish spaces
Let $X$ and $Y$ be two measurable spaces. Let $F:X\rightarrow Y$ be a multifunction. I am familiar with Kuratowski and Ryll-Nardzewski measurable selection theorem which gives a selection of $F$ when $...
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