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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0answers
10 views

connected set as union of three closed which intersect

I'm stuck with the following part in the proof of corollary 4, section C chapter 1, page 22 of the book analytic functions of several complex variables by Gunning and Rossi. Suppose $U \subset ...
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0answers
22 views

to prove that a metric space which is not complete [on hold]

Given $C_{\infty}= \{x=(x_n) :x_n \in \mathbb{R}\ \text{and}\  \exists\ n(x)\in \mathbb{N} $ s.t $x_n = 0\ \forall\ n > n(x)\}$  Where each element is sequence of type ...
1
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0answers
12 views

Regarding metrizability of weak/weak* topology and separability of Banach spaces.

Let $X$ be a Banach space, $X^*$ its dual, $\mathcal{B}$ the unit ball of $X$ and $\mathcal{B}^*$ the unit ball of $X^*$. The following result is well-known Theorem. If $X$ is separable, then ...
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0answers
20 views

Problem about compact subspace of Hilbert cube.

This is my problem: I have already completed part (i), but I really can't see how I can relate compact subspace with homeomorphism in part (ii). Please give me some ideas.
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0answers
6 views

Is a partially topological group completely regular

Let $G$ be a group and $\mathcal T$ be a topology on $G$ and the function $$ \begin{align*} &f:G\times G\to G\\ &f(x,y)=xy^{-1} \end{align*} $$ be continuous at $(1,1)$. Is $(G,\mathcal T)$ ...
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3answers
55 views

Why we need topology to define these new form open sets?

By the definition of topology, I feel topology is just a principle to define "open sets" on a space(in other words, just a tool to expand the conception of open sets so that we can get some new forms ...
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2answers
69 views

Characterization of continuity in terms of preimages of open sets

1--8 Theorem. If $A\subset \mathbb R^n$, a function $f:A\to \mathbb R^m$ is continuous if and only if for every open set $U\subset \mathbb R^m$ there is some open set $V\subset \mathbb R^n$ such ...
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2answers
41 views

Is $\{\langle x,y\rangle\mid 1 \leq x \leq 2, y = 0\}$ compact in $\Bbb R^2$?

Is this set in $\Bbb R^2$ compact: $$\{\langle x,y\rangle\mid 1 \leq x \leq 2, y = 0\}$$ I think it is compact, but the answer says not. Any help is appreciated.
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1answer
38 views

Axioms of seperation

I am studying topological spaces, and I have seen that there are $3$ main axioms of separation: $\mathrm{T1}$, Hausdorff and normal. Now, between Hausdorff and normal there is a case where: given ...
0
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1answer
38 views

open set and cardinality

I've learnt Set Theory, but I didn't learn Topology and Measure Theory. I met a term "open set" today. According to Wiki, An open set is an abstract concept generalizing the idea of an open ...
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1answer
18 views

cells of quotient CW complex

Let $X$ be a CW complex and $Y$ a CW subcomplex. If $X$ has no cell of dimension $n$, for some $n>0$, then $X/Y$ has no cell of dimension $n$. Is it true? Why?
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32 views

topology problem read [on hold]

Not a topology since if we consider $\{x_n\}_{n=1}^{\infty}$ where $x_n = 1- \frac{\sqrt{2}}{2n}$. Clearly, $(x_n \in \mathbb{R}^+ \backslash \mathbb{Q})$, $(\forall n \in \mathbb{N})$. Let $(A_n = ...
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0answers
23 views

Diffeomorphism and hyperbolic points

Suppose $f$ is a diffeomorphism.Prove that all hyperbolic periodic points are isolated. I tried using the mean value theorem using two diferent periodic points (assuming the periodic points arent ...
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1answer
88 views

If $A\times B$ is compact , then $A$ is compact and $B$ is compact?

Is this true? I think so but I can't seem to prove it / know how to. If $A\times B$ is compact then if $(x,y) \in A\times B$ then $x \in A$ and $y\in B$ and $(x,y)$ is covered by finite subsets of a ...
1
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0answers
36 views

Are Borel sets preserved by an open continuous map?

Does an open, continuous function defined on a compact metric space to itself send Borel sets to Borel sets?
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0answers
50 views

Rudin Theorem 2.7

Theorem 2.7 in Rudin's Real and Complex analysis Theorem Suppose $U$ is open in a locally compact Hausdorff space X, $K \subset U$, and $K$ is compact. Then there is an open set $V$ with compact ...
2
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1answer
39 views

Proof of paracompactness of CW-complexes (J. Lee, Introduction to Topological Manifolds)

I have a question about a proof in John Lee's Introduction to Topological Manifolds (5.22). Given CW-complex $X$ with skeletons $X_n$ and open cover $\left(U_\alpha\right)_{\alpha\in A}$, we ...
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2answers
28 views

Homology group of 3-fold sum of projective planes

I want to calculate the homology group of the 3-fold sum of projective planes defined by the labelling scheme $aabbcc$. For this I will use the following corollary from Munkres: Corollary 75.2: Let ...
0
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1answer
31 views

Homology group of space $X$ given by the labelling scheme $aabcb^{-1}c^{-1}$

I have to calculate the homology group of the quotient space $X$ given by the labelling scheme $aabcb^{-1}c^{-1}$ and then determine to which of the following spaces it is homeomorphic: $S^2, ...
0
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1answer
23 views

Arbitrary Fundamental Group and Surfaces

someone had explained to me how to construct arbitrary space $X_G$ such that $\pi_1(X_G) \cong G$, but i don't remember the end. The idea was the following : take a presentation of the group, and ...
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1answer
23 views

The set of all limit points $A'$ of a subset of a topological space $X$ is empty if $\tau = 2^X$

Proposition: If $X$ is a topological space with $\tau = 2^X$, then $A' = \emptyset$ where $A \subset X$ I found the proof and it uses the fact that if $x \in A$, then $\{ x\} \cap A - \{x \} = ...
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2answers
63 views

I neet to prove that the set $ A:=\{ \frac {1}{n} | n \in \mathbb{N}\}\bigcup\{ 0\}$ is closed in R.

$ A:=\{ \frac {1}{n} | n \in \mathbb{N}\}\bigcup\{ 0\}$ is a closed set in $\mathbb{R}$ by the definiton. I can't use that $cl(A)=A$ iff a is a closed set.
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2answers
39 views

Closed sets in product topology

I have an assignment, I have to proof that arbitrary product of close sets is closed in the product topology, I think I have to use complements and treat with opens, what do you think?
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1answer
39 views

Space of Functions: Characterizations of Positivity

Context The problem here is about the characterization of positivity for real or complex valued functions: $$\sigma(f)\geq 0\iff\sigma(f(x))\geq 0\text{ for all }x\in X\iff f(x)\geq 0\text{ for all ...
1
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1answer
27 views

Paradox in connection with definition of limit points and order limit theorem?

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I come across something that appears (to me) as a paradox. Let me first write down one definition and two theorems that ...
2
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0answers
35 views

Basic question about lifting maps to covering spaces

Any continuous map $f: X_1 \to X_2$ "lifts" to a map $\tilde f: \tilde X_1 \to \tilde X_2$ (provided that $X_1$ and $X_2$ have universal covers). The space $\tilde X_1$ is certainly ...
1
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1answer
78 views

Is true the boundary of compact set of $\mathbb{R}^n$ have Measure Zero?

Let $\Omega \subset \mathbb{R}^n$ open and $f:\Omega \rightarrow [0, \infty[$ a measurable function. Suppose that there exist $C>0$ such that $$\int_K f dm < C,\ \forall\ K\subset\Omega,\ K\ ...
1
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1answer
32 views

$X$ is contractible and $Y$ is path connected then $[X,Y]$ has a single point

Im trying to show that: for $X,Y$ topological spaces $X$ is contractible and $Y$ is path connected then $[X,Y]$ has a single point while $[X,Y]$ denote the set of homotopy classes of maps of $X$ ...
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2answers
66 views

How to show that every continuous function from $[0:1]$ to $[0:1]$ has a fixed point?

This exercise is from Munkres topology: Let $f:[0:1]\rightarrow [0:1]$ be a continuous function. How can we prove that there exists some point $x\in [0:1]$ such that, $f(x)=x$? Any ideas please?
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2answers
41 views

Compact Set: Cover by Merely Neighborhoods

Disclaimer: This thread is just a record of thoughts and written in Q&A style. A subset is compact if every open cover admits a finite subcover. What if one replaces open covers with covers by ...
0
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1answer
37 views

Compactness of the Grassmannian $G(k,n)$

Related to this question, suppose we define $G(k,n)$ to be the set of $n\times k$ matricies with rank $k$, equipped with the quotient topology of $\mathbb{R}^{nk}$ by the equivalence relaiton $$A\sim ...
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1answer
54 views

Operators on the family of all subsets of a topological space that maybe generates a base for these family.

I will try to do at least something of my first question. Given a topological space $\langle X,\tau\rangle$, we define two operators on $2^X = \{ A : A \subseteq X \}$ as follows. For $\alpha ...
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6answers
83 views

Rudin's Topological Definition of an Open Set — Does it Disagree with the Metric Space Definition?

I wanted to share this definition of an open set, which made me uncomfortable. It comes from Rudin's Real and Complex Analysis and begins with the definition of a topology: A collection $\tau$ of ...
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1answer
30 views

Homeomorphism of a Genus-2 Surface

Does there exist a homeomorphism from a genus-2 surface, the connected sum of 2 tori, to two circles, $S^1$, intersecting at a point? Intuitively it seems that the double torus can be squeezed into ...
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1answer
60 views

topology defined on the set $\mathbb{R}^\mathbb{R}$?

What is the topology defined on the set $\mathbb{R}^\mathbb{R}$ of functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that allows us to talk about convergence of sequences in $\mathbb{R}^\mathbb{R}$?
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1answer
60 views

Properties of preimages and intersections of sets

I am working through Bert Mendelson's "Introduction to Topology" and am having some trouble with proofs. The text in well presented but to get a proper understanding I am working through the ...
3
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2answers
42 views

Connected topological spaces, product is connected

Show that if $(X_i)_{i \in \mathcal I}$ where $X_i$ is a topological space for every $i \in \mathcal I$, then $X_i$ is connected for every $i$ if and only if $\prod_{i \in \mathcal I} X_i$ is ...
2
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3answers
70 views

Openness of path connected components of open subsets of $\mathbb C$

Let $\Omega\subset \Bbb{C}$ be an open set. My textbook states that every path connected component of $\Omega$ is open. I can't seem to understand why that is. Why does every point have to contained ...
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0answers
49 views

Solutions to Topology by Munkres [on hold]

I was searching for solution to chapters 1, 2 & 3 of the book 'Topology' by James Munkres. Any suggestions where it can be available.
0
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1answer
16 views

Strong approximation of operators.

If I want to approximate strongly an operator $T$ with another in a subset $A \in L(H)$ why is not enough to ask "for every $\epsilon>0$ there is an operator $S\in A$ such that for every $\eta \in ...
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1answer
35 views

Exercise 3.3.8 from Understanding Analysis by Stephen Abbott

Motivation: trying to prove that if $K \subseteq \mathbb{R}$ is compact (and thus, by the Heine-Borel theorem, closed and bounded), then this implies that any open cover for $K$ has a finite subcover. ...
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0answers
37 views

Some questions concerning continuity and relations

A lot of equivalent conditions for functions between topological spaces $$ X\overset f\longrightarrow Y $$ are proved on this site. Here some of them formulated from the perspective of 'relations': ...
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1answer
29 views

Countable product of closed set is closed?

This is my problem: Let $X=\prod_{i=1}^\infty X_i$. Product of $C_i$ closed requires its complement open. i.e. $X\setminus\prod_{i=1}^\infty C_i=\prod_{i=1}^\infty X_i\setminus C_i$ open. But ...
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2answers
21 views

Discussion on Measures: Sigma-Additivity

Disclaimer: Though this thread is written in a Q&A style any new thoughts are really welcome! What reasons are there to restrict measures to countable additivity rather than uncountable ...
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1answer
32 views

Interesting and intuitive affirmation involving convex sets

Let $\Omega_1$ and $\Omega_2$ two open, bounded and convex domains in $R^n$ with $\Omega_1 \supset \overline{\Omega_2}$ and $0 \in \Omega_2.$ Suppose that for each $x_0 \in \partial (\Omega_1 ...
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1answer
26 views

Simple question about an exemple in covers

I don't get the last one (I underlined it in red ) take $(0,1)$(which is an unbounded subset of $\mathbb R$) then if we take $a=0$ then this set $\{(0,1)\}$ will cover the subset $(0,1)$.
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1answer
38 views

metric spaces and topology [on hold]

Let $d_1,d_2$ be metrics on $X$ such that any sequence $(x_n)$ converges in $(X,d_1)$ iff it converges in $(X,d_2)$ to the same point. Must $(X,d_1)$ and $(X,d_2)$ have the same topology?
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0answers
38 views

limit point and topology [on hold]

Can we define a topology on set of naturals $\mathbb N$ in which every point is a limit point? i know that set $\mathbb N$ has no limit point but can we define a topology on $\mathbb N$ such that ...
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1answer
43 views

Continuity definition and theorem in a topology

This is an extremely common theorem, I have a function $f$ that maps $f:(X,\mathscr{S})\to(Y,\mathscr{T})$. I want to show that $f$ is continuous if and only if for all $V\in \mathscr{T}$, ...
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1answer
54 views

Banach Spaces: Totally Bounded vs. Bounded

Are the finite dimensional Banach spaces precisely those ones in which subsets are totally bounded iff they're bounded?