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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.

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I think a definition is wrong in “Model Categories” by Hovey.

I am working through the book "Model Categories", by Mark Hovey, and have a doubt about a definition given there. At the beginning of page 50 we read: Define a map $f:X\rightarrow Y$ to be a closed ...
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Proving that $f:S^1 \to S^1$ with closed degree $n$ is homotopic to the map $z \mapsto z^n$?

Suppose that $f:S^1 \to S^1$ is continuous and has closed degree $n$, how would you show that $f$ is homotopic to the map $z \mapsto z^n$? I know that by definition of closed degree, we have $\deg(f \...
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Products of sites

Does the category of sites (i.e. small categories equipped with a Grothendieck topology) has products? Is there a connection to the product of locales (as discussed in Johnstone's Stone spaces, II.2....
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Topology generated by the subcollections of compact sets of a metric space

For a metric space $(X,d)$ let $\mathbb{K}X$ be the collection of compact subsets of $X$. Give $\mathbb{K}X$ the topology generated by the sets: $$W(U,K)=\{C\in\mathbb{K}X:(K\cup C)-(K\cap C)\subseteq ...
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Do we really need to use the Jordan-Schönflies Theorem to prove that every surface can be triangulated?

I have read that most proofs of the triangulability of surfaces require the use of the Jordan-Schönflies Theorem. However, is such high-tech machinery really needed? The problem is that 3-manifolds ...
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119 views

What is special about simplices, circles, paths and cubes?

There are some ubiquitous families of graphs — the complete graphs (or simplices) $K_n$, the circle graphs $C_n$, the path graphs $P_n$, and the hypercube graphs $Q_n$ — that intuitively ...
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Question about proof of Tychonoff-Alaoglu

I'd like to check that I understand the proof in full detail. Can you tell me if the following is correct? Thanks for your help. Claim: The closed unit ball $B_{\|\cdot\|_{op}}(0,1)$ in $X^\ast$ is ...
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Computing the hypercohomology of a complex of acyclic sheaves

Let $K^{\bullet}$ be a cochain complex of sheaves of finite-dimensional vector spaces, I wanted to compute $\mathbb{H}^{\bullet}(X,K^{\bullet})$ = the hypercohomology of the complex $K^{\bullet}$, the ...
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Existence of measures assigning positive values to all open sets

Let $K$ be a compact Hausdorff space. Does there exist a finite Borel measure on $K$, assigning positive values to all non-empty open sets of $K$?
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Generalizations for Tietze's extension theorem.

Tietze's extension theorem says: ''If $A$ is a closed subset of $X$ a normal space, and $f:A\to \mathbb{R}$ continuous, then we can extend $f$ to a continuous function $g:X\to \mathbb{R}$." I know ...
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tangent space clarification

I was just wondering why we don't define the tangent space of a smooth manifold at a point $p$ to be $\{p\} \times \mathbb{R}^{n}$, rather than using derivations, germs, or equivalence classes of ...
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Homeomorphism groups as topological groups

As is well known, the homeomorphism group of a compact Hausdorff space is a topological group. The same is true for locally compact locally connected Hausdorff spaces, but it is false in general. Now ...
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How to construct certain cover given in Mumford's Abelian Varities book

In chapter I, appendix to section 2 of the book "abelian varieties" by Mumford, we consider a discrete group $G$ acting freely and discontinuously on a good topological space $X$ (i.e., $\forall x \in ...
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What is the $\epsilon^{00}$-topology?

"If $F$ is a locally convex space, then the topology on $F^*$ (the topological dual of F) will be the strong topology, and the topology on $F^{**}$ will be the $\epsilon^{00}$-topology." This is the ...
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Isometries of Hilbert cube

I'm looking for information about the isometries of the Hilbert cube: $Q= \prod_{i=1}^{\infty}[0,1]$, with the distance : $d((x_i),(y_i))= \sum_{i=1}^\infty\frac{|x_i-y_i|}{2^i}$, but I have not ...
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Image of a first category set for a typical continuous function

Given a first category set $A \in [0,1]$, is the set $X = \{f \in C[0,1]:f(A)\text{ is first category}\}$ residual in $C[0,1]$? I tried two strategies: First I tried to play the Banach-Mazur game on $...
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Advanced Math for Reinfrocement Learning - state space and state sequences (policies)

Reinforcement learning has two important notions and I am interested in advanced math that can investigate those notions: State space - set of states. Apparently, deep structures should exist in this ...
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Topology and $\sigma$-algebra on the space of probability measures

Let $(X,\mathbf{A})$ be a $\sigma$-algebra of sets, $\mathscr{M}(\mathbf{A})$ be the set of all $\mathbf{A}$-measurable functions $f\colon X\to [0,1]$, and $\mathscr{P}(\mathbf{A})$ be the set of all ...
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Showing $f(B_r)$ is relatively compact for a certain $f$.

Let $X$ be a Banach space and $g:B_r\to X$ a continuous function, where $B_r:=\{x\in X\mid \|x\|\leq r\}$. Suppose that $g(x)\neq 0$, for all $x\in B_r$. $g(B_r)$ is relatively compact. Define $f:...
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Higher cup-1 product of coboundaries is also a coboundary?

In the cohomology or the group cohomology theory, suppose $\mu_1$ and $\mu_2$ are coboundaries of arbitrary dimensions, $$ \mu_1=\delta \eta_1 $$ $$ \mu_2=\delta \eta_2 $$ where $\eta_1$ and $\eta_2$ ...
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81 views

Is $\mathbb R^J$ normal in the box topology when $J$ is uncountable?

Question: Is $\mathbb R^J$ normal in the box topology when $J$ is uncountable? I know $\mathbb R^J$ is not normal in the product topology, see "Proof" that $\mathbb{R}^J$ is not normal ...
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Diffeotopy group, Mapping Class group, Isometry group

There are several closely related concepts on the symmetries or symmetry groups of the space. My apology, but some vague imprecise definitions may be as: Mapping class group (MCG) is an important ...
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Spivak Calculus on Manifolds - Tangent space on a boundary point of a manifold

I am an undergraduate student who is studying Spivak's calculus on manifolds. I have several questions in the pages 119 and 120 of the book, which are about the tangent space at a boundary point of a ...
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64 views

How many connected components could this set have?

Suppose $U \subset \mathbb R^2$ is an open subset. Let $(x,y)$ denote the coordinates. For any $(x_0, y_0) \in U$, if we fix $x_0$, the set $U_{x_0} = \{y \in \mathbb R: (x_0, y) \in U\}$ has two ...
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193 views

A rigorous yet intuitive summary of inflection and critical points for beginning calculus?

I haven't done these in awhile. While my analysis covered continuity but not differentiability, I have so far not revisited these in learning geometry or algebra. I am trying to help a calculus ...
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If an immersion $X$ maps circles into planes then its image $X(\mathbb{D})$ is homeomorphic to the cylinder.

Let $X:\left( u,v\right)\in \mathbb{D}\backslash \left\{ 0\right\}\subseteq\mathbb{R}^2 \mathbb{% \longmapsto }\left( x\left( u,v\right) ,y\left( u,v\right) ,z\left( u,v\right) \right) \in \mathbb{\...
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Are quotients of topological categories also topological categories?

I am considering quotients of categories as in MacLane's Categories for the working mathematician, as described in the next paragraph. Let $C$ be a (small) category and $R$ an equivalence relation on ...
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A Lipschitz Implicit Function Theorem.

I look for a reference (book or article) that contains the statement of a version of the implicit function theorem as stated below. This statement I found in notes (with due proof) on the implicit ...
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Sets of measure zero and smooth functions

I want to prove: Let $U$ be an open subset of $\mathbb{R}^n$ and $F:U\to \mathbb{R}^n$ a $C^\infty$ function. Let $S$ be a subset of $U$ of zero-measure. Then $F(S)$ has zero-measure. Proof We ...
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Comparing elements of sets

Let $a_1, a_2, a_3, a_4$ be real numbers. Consider the following sets $$ \mathcal{U}_1\equiv \{-a_1, a_1, -a_2, a_2, 0, \infty, -\infty\} $$ $$ \mathcal{U}_2\equiv \{-a_3, a_3, -a_4, a_4, 0, \infty,...
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Stalk at a point of the sheaf of open sets

Fix a base topological space $X$. For every open subset $U$ of $X$, define $\mathcal F(U)$ to be the set of open subsets of $U$. The restriction map $\mathcal F(V) \to \mathcal F(U)$ sends $W$ to $W ...
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Condition that the faces of a graph be unique?

There is a theorem that if a planar graph is 3-vertex-connected, then it has a unique embedding up to a reflection. (See e.g. here.) This means that its faces and its dual graph are uniquely defined. ...
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Searching for examples of “graph algebras”

This question is related to the question https://mathoverflow.net/questions/301626/what-is-known-about-graph-algebras but is not a duplicate, since here I am searching for examples of graph algebras. ...
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Given a Baire one function, is it true that its restriction to special subset continuous?

Background: Let $X$ be a Polish space, that is, a separable completely metrizable topological space. Assume that $F$ is a closed subset of $X.$ Recall that a function $f:X\to\mathbb{R}$ is called ...
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A question on density of double cosets in $\mathrm{SL}_2$

Let $\Gamma$ be a lattice in $G = \text{SL}(2,\mathbb{R})$ and consider the subgroups $$ N^- := \Bigl\{ \begin{pmatrix}1 & 0 \\ x & 1\end{pmatrix} : x \in \mathbb{R}\Bigr\} $$ and $$ A^+ := \...
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What is $\rm cl([-2,2]\times[-3,3])$ in this topology

I have this topology on $\mathbb{R}^2$, $$\sigma=\{\emptyset\}\cup\big\{\bigcup_{r\in A} \Omega_r:~A\subseteq \mathbb{R}^+\big\}$$ where $$\Omega_{r}=\{(x,y)\in\mathbb{R}^2: x^2+y^2=r^2\}$$ I want ...
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special non zero ideals of the ring of continuous functions

Let $R$ be the ring of all continuous real valued functions on a completely regular space $X $, that is $R:=C (X)$, and let $0\not=I $ be an ideal of $ R $ and $\{ I_i \}_{ i\in A } $ be a ...
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Different definition of fibre bundle

I have 2 definition of fibre bundles, one from the book Fibre bundles from Husemoller and the other one from The topology of fibre bundles by Steenrod. And they differ. The first definition says ...
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Soft question - a subset of a Hilbert space endowed with subspace topology

I am considering a Hilbert space $X$, endowed with its weak topology. I need to work with a subset (but not a subspace) $S$ of $X$. However I need to endow $S$ with the subspace topology (so $U$ is ...
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Completeness of Continuous Functions on Uniform Spaces

So I'm trying to find the most general setting in which I can talk about completeness of function spaces. In metric spaces, it's simple to show that the space $C(X,Y)$ of continuous functions $X\...
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Relative volume of $\delta$-fattening (neighborhood) of a compact set

For a non-empty, compact set $A \subseteq \mathbb{R}^n$, the $\delta$-fattening of $A$, $A_\delta$, is defined to be the set $$ A_\delta = \cup_{a \in A} B_{\delta}(a), $$ where $B_\delta(a)$ denotes ...
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Colimits for gluing schemes and the functor of points 1

Closely related questions been asked several times in different forms on here but I feel like none really spell out what's going on. I have been looking more at glueing schemes, and particularly ...
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144 views

deformation retract of contractible spaces

Let $X$ be a topological space and $A$ be a subspace of $X$. A deformation retraction of $X$ onto $A$ is a continuous map $F: X\times [0,1]\longrightarrow X$ such that for any $x\in X$ and any $a\in ...
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Prob. 12, Sec. 23, in Munkres' TOPOLOGY, 2nd ed: If $Y\subset X$, $X, Y$ are connected, and $A, B$ form a separation of $X-Y$,

Here is Prob. 12, Sec. 23, in the book Topology by James R. Munkres, 2nd edition: Let $Y \subset X$; let $X$ and $Y$ be connected. Show that if $A$ and $B$ form a separation of $X-Y$, then $Y \cup ...
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Equivalence of Path-Connectedness and Arc-Connectedness for Hausdorff Spaces

I have a classical sort of question. If we define a path in a space $X$ from points $a$ to $b$ to be a continuous image $f: I \rightarrow X$ from the unit interval to $X$ such that $f(0) = a$ and $f(...
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Munkres Exercise 70.1

This is question number 1 from section 70 (The Seifert-van Kampen Theorem) in Munkres. Assume the hypotheses of the Seifert-van Kampen Theorem. Suppose that the homomorphism $i_*$ induced by ...
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Moral justification for “sheaf=continuously variable set” and local injectivity

From the topos theory perspective it is a general motto that sheaves are continuously variable sets. The internal logic of sheaf toposes justifies this motto, but I would like an additional "local" ...
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Example 4, Sec. 22 in Munkres' TOPOLOGY, 2nd ed: How is this quotient space homeomorphic with $S^2$?

Here is Example 4, Sec. 22, in the book Topology by James R. Munkres, 2nd edition: Let $X$ be the closed unit ball $$ \left\{ \ x \times y \ \vert \ x^2 + y^2 \leq 1 \ \right\} $$ in $\mathbb{...
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Prob. 8 (b), Sec. 20, in Munkres' TOPOLOGY, 2nd ed: All four topology on $\mathbb{R}^\infty$ are distinct

Here is Prob. 8, Sec. 20, in the book Topology by James R. Munkres, 2nd edition: Let $X$ be the subset of $\mathbb{R}^\omega$ consisting of all sequences $x$ such that $\sum x_i^2$ converges. Then ...
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Iterative construction of the real projective space

I can visualize the construction of $\mathrm{RP}^2$ from a disc $B^2$ whose boundary $S^1$ is subjected to the antipodal identification. This can be obtained by glueing the edge of a Möbius strip $M$, ...