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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Finding the interior of a set of continuous functions

Let $C[0,1]$ be the family of continuous functions on $[0,1]$ with the norm $$||x||_{\infty}=\max\limits_{0\leq t\leq 1}|x(t)|.$$ Let $$A=\{x\in C[0,1]|\quad ||x||_{\infty}\leq 1\, x(1)=1\}.$$ Prove ...
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Closure and Boundary of the plane without axes

Consider the plane $\mathbb R ^2$ equipped with Euclidean topology. Remove the axes, that is, $\{(x,y)\in \mathbb R^2 \mid x\neq 0, y\neq 0\}$ and call this subset $M$. What is the closure and ...
Fuze's user avatar
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Prove $(A')' \subset A'$ in general topological spaces

I'm trying to prove the set of accumulation points is closed in general topological spaces, all proofs I've found are for metric spaces, I know arguments are similar but I haven't been able to ...
Boris Tross's user avatar
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Let $\{\vec{x}_k\}_{k=1}^{\infty}$ be convergent, show that $\{\vec{x}_k:k\in \Bbb N\}\cup\{\vec{x}\}$ is a compact set.

Let $\{\vec{x}_k\}_{k=1}^{\infty}$ be a sequence in $\Bbb R^n$ that converges to the vector $\vec{x}\in \Bbb R^n$, then show that $\{\vec{x}_k:k\in \Bbb N\}\cup\{\vec{x}\}$ is a compact set. According ...
Roma_Rayado's user avatar
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The base for the weak topology is actually a base

I am reading Real and Functional Analysis by Serge Lang and I am having problems with this: Let $Y$ be a topological space and let $\mathcal{F}$ be a family of mappings $f:X\rightarrow Y$ of $X$ into $...
Branco Flores Rocha's user avatar
1 vote
1 answer
18 views

Complement of all subdiagonals is open in the infinite product topology

Let $(X,\tau)$ be a Hausdorff space, and consider the countably infinite product $X^\infty$ with its product topology. Question.$\quad$ Is the set $$ X^\infty_\neq := \{ (x_i)_i\in X^\infty : x_i\neq ...
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Weak closure of a subset of the unit sphere of $\ell_1$

It is a well-known and standard fact that for every infinite-dimensional Banach space $E$ the weak closure $\overline{S_E}^w$ of the unit sphere $S_E$ of $E$ is equal to the closed unit ball $B_E$ of $...
Damian Sobota's user avatar
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Proof of limit of vector valued function is equal to vector of limits of each component using open set definition

I know know how the proof of the following theorem is done by using Euclidean norm: Ref: Prove that limit of vector valued function is equal to vector of limits of its component Let $A \subseteq \...
InTheSearchForKnowledge's user avatar
3 votes
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Inverse limits of complete metric spaces is Baire

It is well known that arbitrary products of complete metric spaces are Baire (refer to Dugundji, example). But, what happens when one considers inverse limits of complete metric spaces over an ...
Anupam's user avatar
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Quotient topology after Gluing

Question: Let $X = ([0,2] \times \{0\}) \cup ([0,2]\times \{1\})$ and let $Y$ be the space obtained from gluing each $t$ in the first copy of the interval to the corresponding second copy of the ...
want2know's user avatar
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The compactness of Radon probability measure

Let X be a compact subspace of $\mathbb{R}^n$. Denote by $\mathcal{R}$ the space of Radon measures on X, and $\mathcal{P}$ the space of Radon probability measures on X. In my book, it says that $\...
Lilileaf's user avatar
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What is the difference between free product and direct sum?

This is a sentence I saw in Topology by Munkres."In this section, we study a concept that plays a role for arbitrary groups similar to that played by the direct sum for abelian groups. It is ...
Nushen's user avatar
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Computing homology from finite closed cover [closed]

Consider a finite collection of closed $d$-balls $\{B_1,\ldots,B_n\}$ which cover a smooth $d$-manifold, $M=\bigcup_{i=1}^{n}B_i$. Suppose we wish to compute the (integral) (co)homology of $M$ from ...
rab's user avatar
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Question about uniform convergence in a proof

The below proposition is from David C. Ullrich's "Complex Made Simple" (pages 264-265) Proposition 14.5. Suppose $D$ is a bounded simply connected open set in the plane, and let $\phi: D \...
MathLearner's user avatar
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Monodromy for nonclosed paths

I've came across this proof of a theorem called "monodromy theorem" (in a completely non-analytical context) for closed paths that is as follows: Let $(Y,p)$ be a covering of $X$ and let $\...
ccnptr's user avatar
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Continuity of a function $f: (X,\mathcal T)\to \mathbb{R}$, where $(X,\mathcal T)$ is subspace of $\mathbb{R}$

$(X,\mathcal T)$ is subspace of euclidean space $\mathbb{R}$, where $X=[0,1]\cup[2,3]$ and topology $\mathcal T$ is induced by $\mathbb R$. Show that $f: (X,\mathcal T)\to \mathbb{R}$, where $$f(x)= \...
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Restricted product topology and subspace topology on the ring of Adeles, which is finer?

Let $V := \{ \infty \} \cup \{ p \text{ prime} \}$, we define \begin{equation*} \mathbb{A} := \{ (x_v)_{v \in V} \in \prod_{v \in V} \mathbb{Q}_v \text{ with } x_p \in \mathbb{Z}_p \text{ for ...
The Tralfamadorian's user avatar
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Continuity at a point via closure

I am studying general topology and a question has come to my mind. For a map between topological spaces, the condition that the image of the closure of every subset of the domain is contained in the ...
Amanda Wealth's user avatar
2 votes
1 answer
33 views

Nulhomopty Lemma (Just for a small part)

I was reading the proof for null-homotopy lemma in Munkres's Topology book, and I cannot convince myself for the suggestion: One can replace $S^2$ by the one-point compactification $\mathbb{R}^2 \cup ...
Ulaş's user avatar
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2 votes
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Proving Munkres Theorem 51.3

I am trying to prove Theorem 51.3 from Munkres, that is: Let $f$ be a path in $X$ and let $a_0, \dotsc, a_n$ be numbers such that $0 = a_0 < a_1 < \dotsb < a_n = 1$. Let $f_i\colon I \to X$ ...
JLGL's user avatar
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How to calculate the quasi-component?

Question:X={$0,1$}×{$0$}$∪[0,1] ×${$\frac{1}{n}:n∈\mathbb{N}$}] with the subspace topology $𝒯$ from $(\mathbb{R}^2,\mathbb{T}_{\mathbb{R}^2})$. Find its components and quasi-components. My progress:...
kuanglei's user avatar
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Polar coordinates generate the same σ-algebra as the usual Borel σ-algebra of R^n? [closed]

I want to prove the statement in the title but first some background information: I am studying multivariate probability theory. For a multivariate random variable, one often uses polar coordinates to ...
Victor's user avatar
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1 answer
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When is this closed set compact

Apparently in the polish space $^\omega\omega$ a closed $K\subset\hspace{1mm}^\omega\omega$ is bounded and therefore compact if it is completely below some $f\in \hspace{1mm}^\omega\omega$ as in $K= \{...
L. R.'s user avatar
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Topology of sets defined by real-valued functions (again)

This is a follow-up to this previous question, but a bit more specific. Suppose I have a simple Euclidean space $X = \mathbb{R}$, or $X = \mathbb{R}^2$, or $X = \mathbb{R}^3$, and a continuous real-...
bubba's user avatar
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1 answer
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Does the partition in evenly covered have to finite or not? [closed]

Consider the definition below of evenly covered: $p: E \rightarrow B $ , continuous, surjective. An open set $U \subset B$ is evenly covered by p if the preimage $p^{-1}(U) = \sqcup_{j} V_{j}, V_{j}$ ...
user1300177's user avatar
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Definition of topological quotient

I'm studying general topology and a question has come to my mind. By definition, a continuous map (between topological spaces) is said to be a quotient map if it is surjective and the codomain is ...
Amanda Wealth's user avatar
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1 answer
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Isomorphism between deck transformations and permutations onto the fiber

I came across this problem in Fulton's Algebraic topology textbook (problem 11.39) and I can't seem to get to the bottom of it. The problem is as follows: Let $p:Y\to X$ be a covering, with $Y$ ...
ccnptr's user avatar
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0 answers
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The line with two origins is locally Euclidean, second-countable but not Hausdorff [duplicate]

I am trying to solve the following Problem in the book Introduction to smooth manifolds by Lee: Let $X$ be the set of all points $(x,y) \in \mathbb{R}^2$ such that $y=\pm 1$, and let $M$ be the ...
Philip's user avatar
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1 vote
1 answer
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Armstrong's definition of the boundary of the surface

On page 116 of Armstrong's Basic Topology, the definition of the boundary of the surface is: $x$ belongs to the boundary $\Leftrightarrow $ there exists a neighborhood $U$ of $x$ and a homeomorphism $...
leo's user avatar
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0 answers
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Relationship between the Collatz Conjecture and the Fibonacci Sequence

Once again taking a stab at some Collatz Conjecture related work, using the modified function where odd numbers return $\frac{3n+1}{2}$ and even numbers consider all divisions by 2 as a single step. ...
J. Whidden's user avatar
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0 answers
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Bolzano Weierstrass property is not preserved under product of topological spaces

I have been working on a counterexample for this problem, but I am not sure if this would work, could someone please check it? {Show that, in general, the B-W property is not preserved through the ...
zula medina's user avatar
1 vote
1 answer
56 views

Find some subset of $C[a,b]$ which is open for $\|~\|_\infty$ but not for $\|~\|_1$ [duplicate]

Let $C[a,b]$ be the space of continuous functions on $[a,b]$. For every $x,y\in C[a,b]$, set $$d(x,y)=\sup\limits_{t\in [a,b]}|x(t)-y(t)|$$ $$\rho (x,y)=\int\limits_{a}^{b} |x(t)-y(t)|dt$$ Admit that ...
lee max's user avatar
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2 answers
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Are sets with no limit points not compact?

Let $A$ be some set of some metric space. I am wondering whether it is true that if $A$ has no limit points, then $A$ is not compact? If $A$ has no limit points, then $A$ cannot be sequentially ...
Ray Siplao's user avatar
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order topology is compact if, and only if, X is a complete reticle.

This is my attempt to make the proof, could someone give some advice/correct it? Please I have not learned what a complete reticle is, but doing some research, this is the deffinition I got: $X$ is ...
zula medina's user avatar
2 votes
1 answer
34 views

What is the cardinality of the set of paths in the plane?

What is the cardinality of the set of paths in the plane? In the question Can you explicitly write R^2 as a disjoint union of totally path disconnected sets, Eric Wofsey describes in a comment a ...
kaba's user avatar
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-2 votes
1 answer
45 views

Property of map verification [closed]

Consider the map $f:[0,2]\times [0,1]\rightarrow [0,1]\times [0,1]$ given by $f(a,b)=(a,b)$ if $(a,b) \in [0,1]\times [0,1]$ and $f(a,b)=(2-a,b)$ if $(a,b)\in [1,2]\times [0,1]$. $f$ is continuous ...
monoidaltransform's user avatar
2 votes
1 answer
44 views

Is $z^{t+1}$ a homotopy between $z$ and $z^2$?

I am recently learning about homotopy and mapping degree and I have been very confused over a simple example. Consider the function $z$ mapping from the unit circle $S^1$ in the complex plane to $ \...
Bill's user avatar
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About the proof of Theorem 26 from General Topology by Kelley [closed]

According to the proof of Theorem 26 from General Topology by Kelley, why is the intersection of U[x] and U[u] open?
Pat's user avatar
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2 answers
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Prove or disprove $C:=\{x\in\mathbb{R}^n|f(x)\leq f(x_0)\}$ is closed

Let $f:\mathbb{R}^n\to\mathbb{R}$. Given $x_0\in\mathbb{R}^n$, define $$C:=\{x\in\mathbb{R}^n|f(x)\leq f(x_0)\}.$$ Show that $C$ is closed. My attempt: Let $\{x_n\}$ be a sequence in $C$ that ...
lee max's user avatar
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0 votes
1 answer
24 views

How to show infx∈D⟨p,x⟩>supy∈Ω⟨p,y⟩?

Suppose thet $\Omega \subset \mathbb R^n$ is closed convex, and $ D \subset \mathbb R^n$ is compact convex. If $\Omega \cap D = \varnothing$, then please show$\exists p \in \mathbb R^n$ with $inf_{x \...
Harry's user avatar
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0 votes
2 answers
41 views

Prove the set $U=\{x\in C[0,1]| x(1)\neq x(0)\}$ is not connected

For two functions $x$ and $y$ in $C[0,1]$, define $$d(x,y)=\max\limits_{0\leq t\leq 1}|x(t)-y(t)|.$$Admit that $(C[0,1],d)$ is complete. Set $U=\{x\in C[0,1]| x(1)\neq x(0)\}$ and define $$\varphi: C[...
lee max's user avatar
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0 votes
1 answer
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Simply connected set

An open connected set $S$ in $\mathbb{R}^2$ is said to be simply connected if its complement relative to the whole plane is connected. This definition is mentioned as an equivalent definition in the ...
Mathguide's user avatar
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2 answers
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Are connected components clopen?

I was thinking about this proposition from a book: "The components of $X$ are disjoint nonempty closed subsets whose union is $X$, and thus they form a partition of $X$" And I wondered: ...
some_math_guy's user avatar
4 votes
1 answer
144 views

For which topologies do self-homeomorphisms preserve the collection of open sets?

Inspired by Does a homeomorphism preserve the open sets?, I wanted to ask the corresponding question: while $X$ may have a topology $\mathcal T$ and a homeomorphism from $(X,\mathcal T)$ to $(X,\...
Steven Clontz's user avatar
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30 views

Geometric intuition behind hyper-sphere volume recurrence relation

There is a recurrence relation for calculating the volume of a hyper-sphere and a logical explanation for why it holds. Is there a geometric intuition behind this to help me intuitively understand ...
Hank's user avatar
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4 votes
1 answer
41 views

Example of CW-complex with $G$-action, which is not $G$-CW-complex

Let $G$ be a quasi-compact, Hausdorff topological group and let $G$ act on a CW-complex $X$ such that the $G$-action sends cells to cells and boundaries of cells to boundaries of cells. Further, ...
Fabio Neugebauer's user avatar
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0 answers
30 views

Fundamental Group and Cartesian Product

Could someone please help me with this? I really appreciate any help you can provide. Question: Prove that $\phi : \pi_1(X \times Y, (x, y)) \rightarrow \pi_1(X, x) \times \pi_1(Y, y) : [f] \mapsto ([...
user1300177's user avatar
2 votes
1 answer
41 views

Why is the subgroup of the adelic image $E_{/\mathbb{Q}}$ open?

I am reading Serre's 1972 paper (English) Proposition 22 (24 in English version) where he proves that no elliptic curve over $\mathbb{Q}$ can have surjective adelic image. In the proof he says This ...
Batrachotoxin's user avatar
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1 answer
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Question on Covering Spaces (Topology)

Could someone please help me out with the solution of the following proof (I do not understand the bold and italics part): Let $q : X \to Y$ and $r : Y \to Z$ be covering maps; let $p = r \circ q$. ...
user1300177's user avatar
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Extension of a function from its domain to the closure of its domian in a general topological space

Let $X$ be a topological space and $A \subset X$ be a non-empty subset. Suppose $f: A \rightarrow Y$ is a mapping from $A$ to some finite set $Y$. If $x \in \bar{A}$, can we always find a net $\{x_{\...
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