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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If $X$ is first-countable then a net converges when a subsequence converges?

Let be $X$ and we assume that $(x_\lambda)_{\lambda\in\Lambda}$ is a net such that there exists a cofinal and increasing map $\varphi$ form $\Bbb N$ to $\Lambda$ such that $\big(x_{\varphi(n)}\big)_{n\...
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How the second inequality stands?

Taken from paper "A Universal Law of Robustness via isoperimetry" by Bubeck and Sellke. Theorem 3. Let $\mathcal{F}$ be a class of functions from $\mathbb{R}^{d} \rightarrow \mathbb{R}$ and ...
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Exercise 7(b), Section 31 of Munkres’ Topology

Let $p \colon X \rightarrow Y$ be a closed continuous surjective map such that $p^{-1} \big( \{ y \} \big)$ is compact for each $y \in Y$. (Such a map is called a perfect map.) (b) Show that if $X$ is ...
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Exercise 7(a), Section 31 of Munkres’ Topology

Let $p \colon X \rightarrow Y$ be a closed continuous surjective map such that $p^{-1}\big(\{ y \} \big)$ is compact for each $y \in Y$. (Such a map is called a perfect map.) (a) Show that if $X$ is ...
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Counterexample to one version of Chevalley's theorem?

Chevalley's theorem says that if $f \colon X \to Y$ is a morphism of finite presentation of schemes and $C \subset X$ is constructible, then $f(C)$ is constructible. Here, constructible means $C$ is ...
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Showing that a convex set $A$ is closed if and only if $A \cap \mathcal{l}$ is closed for every line $\mathcal{l} \subseteq \mathbb{R}^{n}$

We assume a convex set $A$ and a line segment $\mathcal{l} \subseteq \mathbb{R}^{n}$. We have to show that $A$ is closed if and only if its intersection with any given line is a closed set. $(\implies)...
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Quotient map on S^1 such that that the quotient is an uncountable space with the indiscrete toplogy.

Based on Leinster Basic Category Theory page 135, problem 5.2.22 (b) I am trying to find a quotient map on the circle $S^1$ which results in the quotient having the indiscrete topology. Using the ...
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1 answer
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The minimality of product topology.

Let $(X,\mathcal O_X), (Y, \mathcal O_Y)$ be two topological spaces, and let $\mathcal B=\{ U\times V \mid U\in \mathcal O_X, V\in \mathcal O_Y \}$. Product topology $\mathcal O$ is a family of sets ...
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Is the surface of a 3d cube homeomorphic to the 2-sphere

Is the surface of a three dimensional cube i.e. $[0,1]^3$ (surface is like a hardboard box, I don't know if it has a symbol to represent) homeomorphic to $S^2$. If yes, is it diffeomorphic to $S^2$ ...
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Why people usually use the term "perfectly normal Hausdorff space" instead of "perfectly normal Fréchet space"?

If a normal T2 space is a normal T1 space and a perfectly normal T2 space is a perfectly normal T1 space, then I would assume people will be more likely to use the latter term because T1 is weaker ...
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Homeomorphism mapping half space to positive orthant

Let $P$ be the positive orthant of $\mathbb{R}^3$ and let $H := \{x_1+x_2+x_3\geq 0\}.$ Is there a way to make a (mostly smooth) map $\phi$ from $H$ to $P$ so that if $k = \{[c,c,c]^T:c\geq0\}$, then ...
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Clarifying the definition of the Tychonoff space $T$ in Fuchs-Fomenko's book

In their topology book, Fuchs and Fomenko define the Tychonoff space $T$ to be the set of all real sequences $(x_1, x_2, x_3, ...)$ with the base of topology formed by the sets $\{(x_1, x_2, x_3, ...) ...
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Proving $|d(x,z)-d(y,z)| \leq d(x,y)$ and $|d(x,y)-d(a,b)| \leq d(x,a) +d(y,b)$ in a metric space $(X,d)$.

Let $(X,d)$ be a metric space, I want to prove the following inequalities: $$\tag{1}|d(x,z)-d(y,z)| \leq d(x,y),$$ and $$\tag{2}|d(x,y)-d(a,b)| \leq d(x,a) +d(y,b).$$ I understand $(1)$ as one side of ...
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Simplicial complex

I started to learn about "simplicial complex" and read about applications but it was very difficult for me to understand these applications, my question is as below what is the importance ...
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Can a, solid, 3-ball be viewed as a Lie group? [duplicate]

I know that the only n-spheres that are Lie groups are $S^0$,$S^1$, and $S^3$. However, if I relax the condition on the points (x,y,z) of the 2-sphere to have a radius less than or equal to one, ...
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Non Locally Compact Space at every point implies Anticompact

Given a topological space $\textbf{(X, $\cal T$)}$, which for convinience we further suposse is Haussdorf. Any finite subset in any given topological space is of course compact. If the space is non ...
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Is the cone over $\mathbb{RP}^3$ manifold? [duplicate]

I am reading "algebraic geometry I, complex algebraic varieties" by David Mumford and having trouble showing the cone over $\mathbb{RP}^3$ is not a manifold. In the last paragraph of section ...
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Can $A(S^-)>A(S^+)$?

Let's be $S$ a surface which is homeomorphic to a $\mathbb{S}^2$ (the unit sphere). Let's be $$S^+=\{p\in S : K(p) \geq 0\} \text{ and } S^-=\{p\in S : K(p) \leq 0\}$$ being $K$ Gaussian curvature. Is ...
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The tent map system is transitive. Can we actually identify any of the infinitely many transitive points, however?

$\newcommand{\O}{\mathcal{O}}\newcommand{\G}{\mathcal{G}}\newcommand{\T}{\mathcal{T}}$TLDR; skip to the end of the preamble - we know that the tent map system is topologically transitive. However, do ...
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Cellular Approximation for Homeomorphism on a Compact Surface.

In my quest to compute the fundamental group of some object in multiple ways, questions concerning cellular approximations of homeomorphisms came up. The setting is as follows: Suppose $M$ is a ...
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Chapter 11, Theorem 5.2 (4) of James Dugundji Topology

Let $p:X \to Y$ be a perfect map. Then: $(4)$ If $X$ is $2^\circ$ countable, so also is $Y$. Dugundji’s proof: Let $\{U_i\}$ be a countable basis for $X$, and let $\{V_i\}$ be the family of all ...
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Problem on finding Intersection form of compact,orientable $4-$manifolds .

$\mathbf {The \ Problem \ is}:$ Let $M$ be an $\mathbb{F}$-oriented manifold of dimension $2 n$ for a field $\mathbb{F}$. Consider the non-singular bilinear form $H^{n}(M ; \mathbb{F}) \otimes H^{n}(M ...
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1 answer
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$f$ is discontinuous. Find topological conditions that there exists a continuous $g$ st. $f(x)\geq0\iff g(x)\geq 0$.

Let $x\in X$ where $X$ is a topological space. It seems like: Claim 1. For every real-valued function $f$ on $X$, the following conditions are equivalent: there exists a continuous real-valued ...
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$\mathbb{R}_l^2$ is not normal. [duplicate]

Let $\mathbb R_l$ be the Sorgenfrey line, i.e. $\mathbb R$ with the lower limit topology. I want to prove that $\mathbb{R}_l\times \mathbb{R}_l$ is not normal. I do not want to use Urysohn's lemma ...
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1 answer
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Show that there exists open sets $P$ and $Q$ in $X$ such that $p \in P, S \subseteq Q$, and $P\cap Q = \emptyset$ in the metric space $(X,d)$

Let $(X,d)$ be a metric space, $S\subseteq X$ is finite, and $p \in X$, but $p \notin S$. Show that there exists open sets $P$ and $Q$ in $X$ such that $p \in P, S \subseteq Q$, and $P \cap Q = \...
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2 answers
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Regarding proof of Bolzano's theorem (Csez Kosniowski)

I am trying to understand the lemma 10.1 (IVT) of "A first course in Algebraic topology" by C. Kosniowski. The lemma states, If $f: I \rightarrow \mathbb R$ continuous with $f(0)f(1) \leq0$, ...
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1 vote
1 answer
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Not a locally compact space that can be represented as union of two locally compact spaces (open and close) [R. Engelking, exercise 3.3.C]

Define a subspace of the real line that can be represented as the union of two locally compact subspaces, one of which is closed and the other open, and that is not a locally compact space.
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1 answer
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$M_k:=\{(x_1,...,x_n)\in \mathbb{R}^n: x_k>x_{k+1}\}$ is open with respect to eucildean metric

I consider the set $$M_k:=\left \{(x_1,...,x_n)^T\in \mathbb{R}^n: x_k>x_{k+1}\right\}\subseteq \mathbb{R}^n$$ for all $ k\in \{1,...,n-1\} $. I tried to show that $ M_k $ is open with respect to $ ...
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2 votes
1 answer
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a continous map from a topological space to itself is open

I was solving some exercises on general topology and the following question came to mind: Let $f$ be a continous map from a topological space $(X,\tau)$ to itself. Is it true that $f$ is an open map? ...
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1 vote
1 answer
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What separation properties are preserved by $f: X \longrightarrow Y$ continuous and surjective

I am asked prove or refute that if $X$ satisfies certain separation axiom and $f: X \longrightarrow Y$ is continuous and surjective then $Y$ also satisfies that certain axiom. For now I have found ...
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-1 votes
1 answer
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Are there other nondegenerate examples of "non-Euclidean" metric topologies?

Note: non-Euclidean here means that the induced topology is not Euclidean, not that the metric itself is non-Euclidean (although this is also a necessary condition). Going off of literature alone, it ...
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1 vote
0 answers
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Interpreting "Infinite regular polygon with infinite sides" as topology

Consider $S$ = a circle, $A[n]$ = regular polygon with $n$ sides, inscribed or circumscribed to $S$. (I think my interpretations work for both situations.) I want to state something like $\lim_{n\to\...
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1 vote
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Are admissible topological rings compactly generated?

Suppose $R$ is a (commutative unital) topological ring which is admissible in the sense of Stacks 07E8: it is complete, Hausdorff, admits a fundamental system of neighborhoods of zero consisting of ...
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Exercise 6, Section 31 of Munkres’ Topology

Let $p \colon X \to Y$ be a closed continuous surjective map. Show that if $X$ is normal, then so is $Y$. [Hint: If $U$ is an open set containing $p^{-1}(\{ y \} )$, show there is a neighborhood $W$ ...
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2 votes
0 answers
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Theorem 5.12 in Schaum's _General Topology_: Topologies characterised by Kuratowski Closure Axioms

How to rigorously prove the following result? Let $X$ be a non-empty set and let $k \colon \mathscr{P}(X) \longrightarrow \mathscr{P}(X)$ satisfy the following Kuratowski Closure Axioms: [K$_1$] $k(\...
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2 votes
0 answers
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Convex combination of closure and interior is in interior of convex set

Let $C\subset \mathbb{R}^n$ a convex set that has nonempty interior. Show: $y\in\mathrm{cl}C,\;x\in\mathrm{Int}C \; \implies\; \lambda x+(1-\lambda)y\in\mathrm{Int}C$ My attempt: The case $y\in\mathrm{...
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2 answers
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When is an open embedding of topological spaces a proper map?

Assume that $f: X\to Y$ is an open embedding. That is, $f$ is a homeomorphism on its image $f(X)$ such that $f(X)$ is open in $Y$. Under what special condition is $f$ a proper map? Thanks.
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1 answer
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Describing the closure and boundary of an open ball for specific epsilon values

Im honestly confused if im understanding this question right at all, but here is what i've come up with so far: so for a. I have with 3 different cases; for elipson = 1, the closure will contain the ...
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2 votes
1 answer
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Tent map is topologically transitive

Let $T:[0,1] \to [0,1]$ be the function $Tx= 2x$, if $ x \in [0, \frac{1}{2}]$ and $Tx = 2-2x$, if $ x \in ( \frac{1}{2} , 1] $. We say that a map is topologically transitive if, for any pair $U, V$ ...
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0 answers
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Prove that the union of two normal and separated subspaces is normal

Let $(X,\tau)$ be a topological space. $A$ and $B$ are separated sets in $(X,\tau)$, i.e.,$\overline{A}\cap B=A\cap\overline{B}=\emptyset$. If the subspaces $(A,\tau\cap A)$ and $(B,\tau\cap B)$ are ...
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1 vote
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Theorem 5.11 in Schaum's _General Topology_: Topologies characterised by neighborhood systems

How to rigorously prove the following result? Let $X$ be a non-empty set and let there be assigned to each point $p \in X$ a class $\mathscr{A}_p$ of subsets of $X$ satisfying the following axioms: [...
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1 answer
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Find the minimum number of colors to color any Map on Torus.

I am finding the minimum number of colors to color any map on torus. I have drawn how complete graph $K_5$ can be embedded on a torus. I know that the chromatic number of this graph is $5$ and we ...
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1 answer
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Question about Homeomorphism and compactness.

I'm currently studying Introduction To Topology and I have two questions I'd appreciate help with! Prove or disprove the following: 1- $(\mathbb{R}, \mathfrak{u}) $ is homeomorphic to $([0,1], \...
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Can I deduce the topology on two sets by knowing the continuous maps between them?

We know that choice of topology on two set is induces a choice of which maps are continuous between two sets. Suppose we knew all the functions between two topological spaces of unknown topology which ...
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1 answer
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Definition of an Open set

So from notes I'm reading, the definiton of an open set is "A subset X ⊂ (a, b) is called open in (a, b) if for every c ∈ X there is an interval (a′, b′) such that (a′, b′) ⊂ X and c ∈ (a′, b′).&...
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1 vote
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Is the following set a continuum?

Let F be a non-empty family of subcontinua of a continuum X such that for any finite subfamily $F_{1},F_{2},...,F_{n}$ in F there is $C\in F$ such that $C \subset F_{1} \cap F_{2} \cap... \cap F_{n}$ ...
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2 votes
0 answers
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Fuchs - Fomenko, Exercise 1.

In their book "Homotopical Topology", Fuchs and Fomenko introduce $\mathbb{R}^\infty$ as the set of all real sequences which have only finitely many non-zero entries, i.e. $\mathbb{R}^\infty ...
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0 votes
1 answer
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How to show that $A_t := \bigg\{u \in \mathbb{R}^n \mid u_1 = 0 \text{ and } \sum_{i=1}^n \lvert u_{i+1}-u_i \rvert \le t \bigg\}$ is closed?

I am having trouble with the following exercise: For $t > 0$ consider the set $$A_t := \bigg\{u \in \mathbb{R}^n \mid u_1 = 0 \text{ and } \sum_{i=1}^n \lvert u_{i+1}-u_i \rvert \le t \bigg\}.$$ ...
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1 vote
0 answers
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Is the following metric space complete?

Let $X=\{x=(x_i)_{i \in \mathbb{N}} \in \mathbb{R}^{\mathbb{N}} \ \vert \ \exists N \in \mathbb{N} : x_i \geq 0 \ \ \forall i \geq N\}$ and let $\bar{\rho}$ be the uniform metric on $\mathbb{R}^{\...
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8 votes
0 answers
106 views

Which rings are rings of continuous functions?

This is a question for which I've found a number of "near-miss" results online, which may actually be answers but whose direct relevance I haven't been able to see. Say that a ring $A$ is ...
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