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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114 views

Two topology questions

1: Let $f\colon S^1 \to \mathbb{R}$ be any continuous map, where $S^1$ is the unit circle in the plane. Let: $$A = \{(x, y) \in S^1 \times S^1 : x\ne y, f(x) = f(y)\}$$ Is $A$ non-empty? If the ...
4
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2answers
129 views

Necessary and sufficient condition for the spaces $\mathbb R/A$ and $\mathbb R/B$ to be homeomorphic

Let $X$ be a topological space. Let $Y$ be a subset of $X$. We denote by $X/Y$ the quotient space of $X$ identifying any two elements of $Y$. Let $A$ and $B$ be two finite subsets of $\mathbb R$. Are ...
7
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1answer
526 views

does every topology have a basis?

This might be a silly question, but i was wondering, is there any topology that cannot be generated by a basis? if not, given a topology, is there a reliable way of figuring out a basis for it? it ...
4
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3answers
122 views

Describe a necessary and sufficient condition for the spaces $\mathbb R\setminus A$ and $\mathbb R\setminus B$ to be homeomorphic.

Let A and B be two finite subsets of $\mathbb R$. Describe a necessary and sufficient condition for the spaces $\mathbb R\setminus A$ and $\mathbb R\setminus B$ to be homeomorphic. I think $|A|=|B|$. ...
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2answers
102 views

Using the limit point property in metric spaces to prove existence of maximum distance.

Let $M$ be a metric space and suppose that $K \subset M$ is a non-empty compact set. So if $p$ is any element of $M$, then there is a point $q$ that belongs to $K$, such that $d(p,x)\leq d(p,q)$ for ...
0
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1answer
57 views

Paracompactness and Filters

If one use ultrafilters to describe a compact space, one gets Tychonoff Theorem as a trivial result. So, im just asking if there is such a "useful" equivalence, but concerning paracompact spaces ( ...
6
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2answers
134 views

Completeness and Topological Equivalence

How can I show that if a metric is complete in every other metric topologically equivalent to it , then the given metric is compact ? Any help will be appreciated .
1
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1answer
48 views

Non Pseudocompact Spaces and C-embedded sets

How can I prove that if a Tychonoff space is not Pseudocompact then it contains a countable infinite c-embedded subset? What we can say for the space which is not countably compact. Of course it will ...
2
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1answer
105 views

Continuous invariants of topological spaces

I would like some clarification on exactly what a 'continuous invariant' of topological spaces is. My book does not give a straight definition but rather just says "Properties preserved by continuous ...
6
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1answer
142 views

About Countable Basis

Is there any "second-countable" theorem ? With this i mean if there is any result like Nagata-Smirnov Theorem (that states necessary and sufficient condition for a space be metrizable), but for ...
9
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1answer
330 views

Topology induced by the completion of a topological group

Let $G$ be an abelian topological group and let $\hat{G}$ be its completion, i.e. the group containing the equivalence classes of all Cauchy sequences of $G$. What exactly is the topology of ...
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0answers
60 views

Pointfree generalization of uniform spaces?

Topological spaces generalize as frames and locales. But are there a pointfree generalization of uniform spaces?
1
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1answer
205 views

where can i find this A.H. Stone's theorem proof?

can someone tell me where can i find a proof of the following theorem (by A.H.Stone) : "an uncountable product of Hausdorff non-compact spaces is never normal " ? thanks in advance !
6
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1answer
464 views

clarification errata in Munkres Topology?

While reading the second edition of Munkres' Topology, I came across this (page 129): Theorem 21.1 Let $f: X \rightarrow Y$; let $X$ and $Y$ be metrizable with metrics $d_X$ and $d_Y$, ...
2
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1answer
269 views

Help in metric space [duplicate]

Possible Duplicate: An open ball is an open set How to prove that in any metric space an epsilon-neighborhood is an open set? solution: Suppose x belongs to V(p). Then d(x, p) Not every ...
3
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4answers
218 views

A question about a closed set

Let $X = C([0; 1])$. For all $f, g \in X$, we define the metric $d$ by $d(f; g) = \sup_x |f(x) - g(x)|$. Show that $S := \{ f\in X : f(0) = 0 \}$ is closed in $(X; d)$. I am trying to show that $X ...
0
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1answer
361 views

Let $S^1$ denote the unit circle in the plane $\Bbb R^2$. Pick out the true statement(s)

Let $S^1$ denote the unit circle in the plane $\Bbb R^2$. Pick out the true statement(s): (a) There exists $f : S^1 \to\Bbb R$ which is continuous and one-one. (b) For every continuous function $f ...
2
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1answer
112 views

Difference between $\mathbb{R}^{\mathbb{R}}$ and $\mathbb{R}^{\omega_1}$

This question is a follow up on my comment-question in this thread. It appears that there was some resolution in the end, but I would still like to know more about this. An internet search turned up ...
3
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1answer
803 views

Homeomorphism that maps a closed set to an open set?

In my Real Analysis class I got a bit frisky and broke out a homeomorphism in a problem to show that a set was closed (that is, I had a closed set, and I made a homeomorphism between it and the set in ...
6
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1answer
328 views

$\mathbb{R}^\mathbb{R}$ is not normal

Does anyone know how to prove that $\mathbb{R}^\mathbb{R}$ (with the product topology) does not fulfill the $T_4$ axiom? It would be sufficient to have an uncountable subset $A \subseteq ...
2
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1answer
90 views

Let $X$ be a topological space, $A\subseteq X$ and $D(A)$ the boundary of $A$…

Let $X$ be a topological space and let $A \subseteq X$. Let $D(A)$ denote the boundary of $A$, i.e. the set of points in the closure of $A$ which are not in the interior of $A$. A closed set is ...
6
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2answers
417 views

Why can't a neighborhood be a finite set?

Rudin defines a neighborhood as follows: Let $X$ be a metric space endowed with a distance function $d$. A neighborhood of a point $p \in X$ is a set $N_r(p)$ consisting of all $q \in X$ such that ...
4
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1answer
150 views

Sum of Cauchy sequences is Cauchy in an Abelian Topological Group

Let $G$ be a topological abelian group and suppose $0$ has a countable fundamental system of neighborhoods. Let $(x_n),(y_n)$ be Cauchy sequences of $G$. Why is it true that $(x_n+y_n)$ is a Cauchy ...
2
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2answers
104 views

Question regarding a remark in Hatcher's Algebraic Topology

I am having difficulty understanding the following remark made by Hatcher on page 50 of Algebraic Topology: My understanding of this paragraph is as follows: For a basepoint $s_0\in S^1$ the ...
0
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1answer
225 views

Preimage of a continous surjection from (0,1) to [0,1].

For a continuous function $f: X \to Y$, the preimage of every closed set in $Y$ is closed in $X$. Let $g: (0,1) \to [0,1]$ be a continuous surjection. Isn't the preimage of $[0,1]$ = $(0,1)$ open?
2
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2answers
154 views

True statements for a continuous function

Let $f\colon \mathbb R\rightarrow \mathbb R$ be a continuous function. Define $G = \{(x, f(x)) : x \in \mathbb R\} \subseteq \mathbb R^2$. Pick out the true statements: a. $G$ is closed in $\mathbb ...
2
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1answer
131 views

Lebesgue's topological dimension

I was reading the definition of dimension from the book: "Topology", Munkres, 2nd ed., and surely I didn't understand, but I wonder how $\mathbb{R}^2$ can have dimension 2. Take the open sets ...
3
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1answer
272 views

Are all compact sets in $ \Bbb R^n$, $G_\delta$ sets?

Are all compact sets in $\Bbb R^n$, $G_\delta$ sets? I know that compact set is bounded and closed.
3
votes
1answer
318 views

Prove that the baker map $B(x)$ is chaotic.

I would like to show that $B(x)= 2x$ if $\ 0\leq x\leq 1/2$ $B(x)=2x-1$ if $\ 1/2 \leq x \leq 1$ is chaotic on [0,1]. I used the symbolic dynamics; any suggestions please? I briefly recall some ...
3
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2answers
3k views

Prove: Every compact metric space is separable

How to prove that Every compact metric space is separable$?$ Thanks in advance!!
3
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2answers
478 views

Every point closed $\stackrel{?}{\Rightarrow}$ space is Hausdorff

If a topological space is Hausdorff, then every point is closed. Is the converse true? Edited: Let $G$ be a topological group and $H$ the intersection of all neighborhoods of zero. Since every coset ...
5
votes
2answers
223 views

Question about the Souslin property

I've been working on problems dealing with the Souslin property. A topological space $X$ has the Souslin property if every pairwise disjoint family of non-empty open subsets of $X$ is countable. I ...
4
votes
2answers
610 views

Exercise 1.1.18 in Hatcher's Algebraic Topology

Background I am currently trying to solve exercise 1.1.18 in Hatcher's Algebraic Topology. The part of the exercise I am interested in is the following: Using the technique in the proof of ...
3
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2answers
190 views

Is it true that every normal countable topological space is metrizable?

I've been reading about and working on various proofs about metrizabililty. I'm having trouble answering the following question: Is it true that every normal countable topological space is metrizable? ...
4
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1answer
99 views

A question about non-compact metric spaces.

I noticed the following statement on a comprehensive exam, and I am having trouble proving it. Can anyone help? If $X$ is a non-compact metric space, then there exists a continuous function $f: X ...
2
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1answer
53 views

complex projective space with some identification with groups

Could any one tell me how to show : $\mathbb{C}P^n ≅ SU ( n + 1)/ U ( n)$? $\mathbb{C}P^n ≅ S^{2 n +1}/ U (1)$ ? what is transitive group of $\mathbb{C}P^n$?
3
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2answers
152 views

Why the weak * topology on the dual of a Banach space has the stronger meaning of locally compact

Let us say that for a Hausdorff topological space to be locally compact means that every point has a compact neighborhood. Why do locally compact have the property that if $x \in U$ and $U$ is open ...
5
votes
1answer
82 views

A property dealing with complete metric spaces

I came across a property in a textbook that caught my eye. The property is: If $X$ is a complete metric space, then the intersection of any two dense $G_{\delta}$-subsets of $X$ is dense in $X$. This ...
1
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1answer
130 views

A condition for the image of a metrizable space to be metrizable

I noticed this problem on a previous exam that I completely missed, and I was wondering if anyone could help me out. Suppose $f: Y \rightarrow X$ is a continuous mapping of a separable metric ...
3
votes
1answer
125 views

Interesting question about the Sorgenfrey line

I was looking over previous comprehensive exams, and I saw this question about the Sorgenfrey line: Can the Sorgenfrey line be represented as the union of three metrizable subspaces? I haven't ...
3
votes
1answer
101 views

Properties of $C_{p}([0,1])$

Earlier this year, thanks to various users on this site, I was able to answer a question dealing with the properties of $C_{p}([0,1])$ here: Properties of Cp(X). $C_{p}([0,1])$ is the space of all ...
5
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2answers
189 views

Question about the cardinality of a space

I've been having conflicting thoughts about the following problem, and I was wondering if anyone could help me out. Is is true that the cardinality of every regular separable space does not ...
5
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1answer
146 views

Would this space be homeomorphic to the set of irrationals?

I've been reviewing various problems dealing with interesting homeomorphisms, and I came across this one. Is the product of the space of irrationals and the space of rationals homeomorphic to the ...
4
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2answers
352 views

Determining whether or not spaces are separable

I've been going over practice problems, and I ran into this one. I was wondering if anyone could help me out with the following problem. Let $X$ be a metric space of all bounded sequences $(a_n) ...
11
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1answer
176 views

Must a space homeomorphic to $\Bbb R\setminus \Bbb Q$ have a countable complement?

There is a problem from a list suggested practice problems that I am having issues with. It says: Suppose that $X$ is a subspace of the real line $\mathbb{R}$ which is homeomorphic to the space of ...
4
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1answer
104 views

Every paracompact space with the Suslin property is Lindelöf

I've been reviewing exams, and I came across this problem that I am having trouble with. Can anyone help me out? Show that every paracompact space with the Suslin property is Lindelöf. A ...
1
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2answers
180 views

Characterization of the Closure of a Set

Say I have a metric space $(X, d)$ and a set $A\in X$. I want to prove that if $a \in \overline{A}$ then there is a convergent sequence $\{x_n\}$ that converges to $a$. Could I have any help?
2
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3answers
731 views

the cone is contractible

Let $X$ be a topological space. I want to show that the cone $CX$ is contractible. Here we construct a deformation retraction from $CX$ to the tip point of the cone $$H_t: CX\to CX;\; (x,t')\mapsto ...
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1answer
88 views

What's the meaning of $C$-embedded?

What's the meaning of $C$-embedded? It is a topological notion. Thanks ahead.
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4answers
173 views

Three questions about fixed points

Pick out the true statements. Let $f : [0, 2] \to [0, 1]$ be a continuous function. Then, there always exists $x \in [0, 1]$ such that $f(x) = x$. Let $f : [0, 1] \to [0, 1]$ be a continuous ...