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; ...

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1answer
106 views

A possible contradiction to “A $T_{1}$ space is countably compact iff every countable family of closed sets has a nonempty intersection”?

My textbook says "A $T_{1}$-space is countably compact iff every countable family of closed sets having the finite intersection property has a non-empty intersection" (Principles of General Topology, ...
4
votes
1answer
62 views

Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function and $A \subseteq \mathbb{R}$

Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function and $A \subseteq \mathbb{R}$ (i) If A is connected, is $f^ {−1} (A)$ so? (ii) If A is compact, is $f^{−1} (A)$ so? (iii) If A is finite, ...
4
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4answers
556 views

Finding a counterexample; quotient maps and subspaces

Let $X$ and $Y$ be two topological spaces and $p: X\to Y$ be a quotient map. If $A$ is a subspace of $X$, then the map $q:A\to p(A)$ obtained by restricting $p$ need not be a quotient map. Could you ...
1
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1answer
90 views

How to prove this fundamental relationship $ b=\ell+n-1$?

How to prove this fundamental relationship? In a network or circuit, number of loop, nodes and branches has to satisfy the following fundamental relationship: $$ b=\ell+n-1,$$ ...
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0answers
48 views

Topological graphs

Given the universel covering space $\hat{X}$ of $X$ by $p:\hat{X}\rightarrow X$, there exists a bijection between subgroups $H<G=\pi_1(X,x_0)$ and covering spaces $\tilde{X}\rightarrow X$ with ...
2
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3answers
157 views

any open ball of radius $2$ is an infinite set?

Is it true that in an infinite metric space, any open ball of radius $2$ is an infinite set? for example $\mathbb{R}^2$ with discrete metric we have $d(x,y)=1\forall x\ne y$ so in this case also we ...
3
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4answers
87 views

Is the set $\{(x, y) : 3x^2 − 2y^ 2 + 3y = 1\}$ connected?

Is the set $\{(x, y)\in\mathbb{R}^2 : 3x^2 − 2y^ 2 + 3y = 1\}$ connected? I have checked that it is an hyperbola, hence disconnected am i right?
0
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1answer
122 views

$[0, ∞)$ onto the unit circle which is not a homeomorphism

Give an example of a continuous map of $[0, ∞)$ onto the unit circle which is not a homeomorphism. $f(x)=e^{ix}$ works?
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2answers
941 views

Proving derived sets are closed

I am following a proof of the statement The derived set(the set of accumulation points) $A'$ of an arbitrary subset $A$ of $\mathbb{R}^2$ is closed. in a book. It starts with Let $q$ be a ...
2
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2answers
86 views

Projections Open but not closed [duplicate]

I often read that projections are Open but generally not closed. Unfortunately I do not have a counterexample for not closed available. Does anybody of you guys have?
1
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2answers
96 views

morphisms on topological spaces

In the category of topological spaces: 1.) Show that a morphism is monic IFF it is injective 2.) Show that a morphism is epic IFF it is surjective 3.) Are there any morphisms that are monic and ...
3
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1answer
143 views

Tietze–Urysohn's lemma in $\mathbb{R}^n$

Let $F_1$ and $F_0$ be closed subsets in $\mathbb{R}^n$, $F_0\cap F_1=\varnothing$. How to build a $C^{\infty}$- function $f:\mathbb{R}^n\to \mathbb{R}$, such that $f|_{F_1}=1$, $f|_{F_0}=0$ and ...
2
votes
1answer
149 views

Proper map on from compact manifolds

I'm stuck on this statement. Could anyone please help me out? Let $X$ be a compact manifold, every map $f: X \longrightarrow Y$ is proper. The definition of proper: a smooth map between manifolds is ...
0
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1answer
63 views

How could we see that a pseudocomplete space is always pseudocompact?

A space $X$ is called pseudocomplete if it has a sequence $\{\mathcal B_n:n \in \omega\}$ of $\mathcal{\pi}$-bases such that for any family $\{B_n: n\in \omega\}$ with $B_n\in \mathcal B_n$ and ...
2
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2answers
78 views

Proving every open subset $H$ of the plane $\mathbb{R^2}$ is the union of open discs

I am going through the proof of Every open subset $H$ of the plane $\mathbb{R^2}$ is the union of open discs in a book (open disc is the standard open Euclidean disc or open ball). It goes ...
2
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2answers
147 views

Terminologies related to “compact?”

A set can be either open or closed, and there can either be a finite or infinite number of them. A "compact" set is one where every open cover has finite subcover. Is there such a thing as a set ...
2
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2answers
476 views

Do all the open sets containing a limit point of an infinite countably compact subset have to contain infinite points?

Say an infinite set is countably compact (if set $E$ is an infinite countably compact set, it contains at least one limit point within itself). Let $x$ be one such limit point of $E$. My textbook says ...
2
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0answers
51 views

field lines terminating at infinity

A dipole consists of two equal and opposite point charges separated by a fixed distance. With two exceptions, all the electric field lines begin on one charge and end on the other. In the two ...
7
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2answers
163 views

Complement of a Topology

Let $(X, \tau)$ topology, I was wondering, if given $$\tau'=\{A^C \mid A \in \tau\}$$ did $\tau'$ also a topology on $X$? If so, why? Thank you.
3
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1answer
466 views

Prove that a standard torus is diffeomorphic to $ \mathbb S^1\times \mathbb S^1$

I was asked to prove that a standard torus(which means we don't consider those pathological cases where it intersects with itself, e.g horn torus) is diffeomorphic to $ \mathbb S^1\times \mathbb S^1$. ...
0
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2answers
85 views

Bounded partial quotients set is nowhere dense

I've stumbled upon a claim that the set: $$ B_N = \{[a_0;a_1,a_2,...] | \exists n_0 >0\forall n\geq n_0 a_n<N\} $$ for some $N$, is nowhere dense (and closed). Unfortunately, I have found that ...
4
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0answers
142 views

Locally connected and compact Hausdorff space invariant of continuous mappings

I am looking for a reference (not proof) to the following theorem: If $X$ is a compact and locally connected topological space, Y is a Hausdorff topological space, $f:X\to Y$ is continuous and ...
5
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3answers
291 views

Homomorphism/map in both direction implies isomorphism/homeomorphism or not?

I was working on a homework, and my first attempt get me to a deadend, but I was eventually able to solve it using a different method. But the fail attempt make me curious, and I wonder if it could ...
2
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1answer
197 views

Notation for set of all closed sets

Is there a common notation for the set of all closed sets of a topological space? I have been using $(X,\tau)$ to denote a topological space with $\tau$ being the topology, set of all open sets. I ...
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2answers
91 views

Applying the contra positive of the finite intersection property

I'm reading a proof which has the following setting. I have a family $D$ of compact sets with empty intersection. The next line takes a finite subset of $D$ with empty intersection. This is ...
0
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1answer
56 views

Isomorphism of Covers

On page 26 of Peter May's A Concise Course on Algebraic Topology, it is claimed that given any two covers of a space $X$, $(E, p)$ and $(E', p')$ are isomorphic iff for any points $e \in E, e' \in E'$ ...
3
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0answers
109 views

Notation $X/Y$ in Vickers 'Topology via Logic

I am currently working through Vickers' book 'Topology via Logic'. At one point (Proof of Theorem 4.4.2) he uses the notation $X/Y = \{z\in S \mid \forall y\in Y: y\wedge z\in X\}$ (and $X\wedge Y = ...
1
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1answer
43 views

help me in trace of following proposition

In a paper an author proved the following proposition Please help me in trace proof of following proposition Proposition: let $f$ be a homeomorphism of a connected topological manifold $M$ with ...
3
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1answer
43 views

Brouwer transformation plane theorem

Can somebody show that BPTT, version 2 is deduced from BPTT, version 1 [BPTT, version 1] Let $h$ be a fixed point free orientation preserving homeomorphism of $\mathbb{R}^2$ Every point ...
6
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1answer
104 views

periodic point of homeomorphism of plane

Let $h$ be a homeomorphism of $\mathbb{R}^2$ onto itself such that $h(K)=K$ for some compact subset $K$ of $\mathbb{R}^2$. Show that $h$ has a periodic point in $\mathbb{R}^2$.
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2answers
247 views

CW-pairs are good pairs

Hatcher uses in a proof that every subcomplex of a CW-complex is a deformation retract of some neighborhood. In what way can I see this in the infinite dimensional case?
2
votes
1answer
180 views

Proof of an equivalent definition of a continuous function

In Dudley, Real Analysis and Probability (2nd ed.), Theorem 2.1.2 states: Given topological spaces $(X,\mathcal{T})$ and $(Y,\mathcal{U})$ and a function $f:X\to Y$, if for every convergent filter ...
0
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1answer
73 views

Could someone help me to improve the proof writing?

I will prove the following claim. I'm not a native English speaker. Could someone help me to improve the writing? A regular pseudocompact Moore space is ccc and first countable. Prove: I will ...
7
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2answers
574 views

Two spaces homotopy equivalent to eachother, attaching maps, Algebraic Topology.

I have a question regarding algebraic topology with which I was hoping someone could help me with. I've managed to show the following: If $f,g:S^{n-1} \to X$ are homotopic maps, then $X\sqcup_fD^n$ ...
0
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1answer
96 views

What is the most appropriate way to cite others' results as a Lemma?

Recently I am writing something up for possible publication. I am struggling with this question: What is the most appropriate way to cite others' results as a Lemma? For example, this is a ...
0
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1answer
69 views

Prove $SO(3)$, the group of rotations of $\mathbb{R}^3$, is not homotopically equivalent to $S^1\times S^2$

Prove $SO(3)$, the group of rotations of $\mathbb{R}^3$, is not homotopically equivalent to $S^1\times S^2$. I know that $\pi_1(SO(n))\cong \mathbb{Z}_2$ and I think that $P:\mathbb{R}\times S^2 \to ...
1
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2answers
113 views

basic doubt about topological manifold

In his book "Introduction to Smooth Manifolds", J.M. Lee defines a topological manifold to be a second countable, Hausdorff space with every point having a neighbourhood homeomorphic to an open subset ...
0
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0answers
72 views

How to construct weak-star convergence?

From partial derivative as vector basis $\left\{\dfrac{\partial f}{\partial x_i}, i=1,\ldots,n\right\}$. How to contruct sequence $\{u_i\}, i=1,\ldots, n$ such that: 1, $u_i \stackrel{w^*}\rightarrow ...
1
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0answers
84 views

Why is the pullback of a connected cover not necessarily connected?

In particular, I read somewhere that the fiber product of the maps $S^1\rightarrow S^1$ sending $z\mapsto z^m$ and $S^1\rightarrow S^1$ sending $z\mapsto z^n$ is disconnected with $\gcd(n,m)$ ...
2
votes
1answer
99 views

Boundary of Closure

Let $X$ be a topological space. I know it is not generally true $\partial A=\partial{\overline {A}}$ for every $A\subset X$. I found it is true if $A$ is a closed ($A=\overline A)$or regularly open ...
24
votes
1answer
833 views

Is the image of a nowhere dense closed subset of $[0,1]$ under a differentiable map still nowhere dense?

Let $f:[0,1]\to[0,1]$ be a continuous function such that its derivative $f'$ exists on $(0,1)$. My question is: Q1. If $E\subset[0,1]$ is a nowhere dense closed subset, is $f(E)$ also nowhere ...
0
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2answers
145 views

little problem about open set in the definition of topology

Definition 1 Let $X$ be a set of points. A collection of subsets $U = \left\{U_{\alpha }\right\}$ forms a topology on $X$ if Any arbitrary union of the $U_{\alpha }$ is another set in the ...
15
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1answer
167 views

Convex metric on a contiuum.

I am having troubles with this problem. Translation: A metric $d$ on a continuum is called convex if for any two points $p$ and $q$ of $X$ there is a point $x\in X$ such that ...
1
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1answer
85 views

Continuity of a level-sets mapping

Let $f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ be a continuous function. Define the set-valued mapping $S: \mathbb{R}^n \rightrightarrows \mathbb{R}^m$ as $$S(x) := \left\{ y \in ...
7
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1answer
285 views

Is $C^k(X,Y)$ a Lindelöf space?

Let $X$ be a compact, finite dimensional smooth manifold and $k\in \mathbb{N}$. Consider the following two cases: $Y$ is a finite dimensional manifold $Y$ is an infinite dimensional Banach manifold ...
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2answers
83 views

Does $X$ have countable network if it has countable extent?

Let $X$ have a $\sigma$-discrete network and have countable extent. Does $X$ have countable network? A family $\mathcal N$ of subsets of a topological space $X$ is a network for $X$ if for ...
3
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1answer
73 views

The Souslin number of every regular perfect pseudocompact space is countable

Here are two questions while I am reading a paper of Arhangelskii's: How could we see that the Souslin number of every regular perfect pseudocompact space is countable? Why is every Moore space a ...
4
votes
1answer
68 views

Must $X$ be lindelöf if it has countable network?

A network is like a base, except that its members need not be open sets. A family $\mathcal N$ of subsets of a topological space $X$ is a network for $X$ if for every point $x\in X$ and any ...
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2answers
285 views

Every bounded non countable subset of $\mathbb{R}$ has a bothsided accumulation point.

Inspired on proving that every compact set of the Sorgenfrey line is countable. Trying to prove any of these in $\mathbb{R}$: 1) Every bounded non countable set has a both-sided accumulation point. ...
4
votes
3answers
91 views

Open neighborhoods of a $G_\delta$ set

This may have a simple answer, but I couldn't find it so far either in textbooks or in math.stackexchange. Let $X$ be a metric space, and $$A=\bigcap^\infty_{n=1}A_n$$ a $G_\delta$ subset of $X$, ...