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
36 views

Compact metrizable space is separable proof question.

Prove that a compact metrizable space is separable. I am confused by a specific case of a compact metrizable space. Let $[0,1]$ be a compact metrizable space. Since $[0,1]$ is metrizable, it's ...
2
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1answer
62 views

if $M$ is compact, then every continuous bijection $F:M\to N$ is an homeomorphism

My book proves that: if $M$ is compact, then every continuous bijection $f:M\to N$ is an homeomorphism by the following: Being $f$ closed, your inverse $g:N\to M$ is a function such that $F\subset ...
1
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1answer
33 views

elements of topology is open set?

Let's consider the topological space $(\mathbb{R},2^\mathbb{R})$, then each interval is element of topology $2^\mathbb{R}$, but as usually, we consider $[a,b]$ as as closed set. So my question is ...
3
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0answers
39 views

Prove $\int_\Omega f(x) \,dx=f(x_B) \int_\Omega1 dx+ \mathcal O(\int_\Omega1 dx \cdot \sup_{x,y\in\Omega}\|x-y\|_2^2)$?

Let $\Omega \subset \Bbb R^n$ be a convex domain and $f: \Omega \to \Bbb R $ and $f \in \mathcal C^2(\Omega)$. Let $x_B $ be the barycentre of $\Omega$ with $$x_B:= \frac{\int_\Omega x \,dx}{\int_\...
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0answers
15 views

A continuum X is hereditarily indecomposable iff C(X) is uniquely arcwise connected

I have to prove as for my homework that a continuum $X$ is hereditarily indecomposable iff $C(X)$ is uniquely arcwise connected, where $C(X)$ denotes the hyperspace of all subcontinua of $X$. Here's ...
2
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2answers
33 views

two metrics on X such that lim d1(xn,x)=0 <=> lim d2(xn,x)=0, does it imply the identity of the two induced topologies?

Two metrics $d_1, d_2$ on $X$ For all $x_n, x$ from $X$ it holds: $$\lim d_1(x_n,x)=0 \iff \lim d_2(x_n,x)=0$$ Does it imply that the topology induced by $d_1$ is the same as the topology induced by $...
1
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0answers
29 views

Question about proof ot Tychonoff's theorem for metric spaces

Tychonoff's theorem: The cartesian product $M = \prod_{i=1}^{\infty}M_i$ is compact $\iff$ each $M_i$ is compact. My book, before proving it, says that the proof will happen like this: Given an ...
3
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2answers
38 views

2-Sphere is connected

I am supposed to show, that if I have a continuous function $F:S^2→ (\mathbb{R}, | · |)$, with $S^2$ being the 2-Sphere in $\mathbb{R^3}$, that there is a point where $F(x)=F(-x)$. Now I know how to ...
2
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2answers
31 views

Question about Definition of homeomorphism (counter example)

I'm teaching my self topology with the aid of a book, but i'm confused about the meaning of homeomorphic. Below, I have 2 topologies, $\mathscr{T}_1$ and $\mathscr{T}_2$ and I'm pretty sure they are ...
0
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2answers
65 views

$M\times N$ compact $\implies$ $M$ compact and $N$ compact

I must prove that $M\times N$ compact $\implies$ $M$ compact and $N$ compact using the definition that, if a metric space $M$ is compact, then every cover has an open finite sub cover. $$M=\cup ...
1
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1answer
20 views

Showing a space is homeomorphic to the Hilbert cube.

Let $(X_i,T_i)$ be a countably infinite family of topological spaces each of which is homeomorphic to the Hilbert cube. Show that $\prod_{i=1}^{\infty}(X_i,T_i) \cong I^{\infty}$. The question also ...
2
votes
2answers
47 views

Theorem 2.17 from RCA Rudin

I understood the proof of points $(a)$ and $(c)$. But I can't understand the proof of $(b)$. It's obvious that every closed set is $\sigma$-compact. But how Rudin applies $(a)$ here? We have to show ...
2
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1answer
39 views

Is $(X,\mathcal T)$ a $T_0$-space?

Let $(X,\mathcal T_1)$ and $(X,\mathcal T_2)$ be topological spaces. Now define $\mathcal T=\mathcal T_1 \cap \mathcal T_2$. If $(X,\mathcal T_1)$ and $(X,\mathcal T_2)$ are $T_1$-spaces, is $(...
0
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2answers
30 views

Proving this quotient space is a Hausdorff space

Define $S^1 = \{x \in \mathbb{R}^2 : x^2 + y^2 = 1 \}$. Define the equivalence relation $\sim$ as follows: $(x,y) \sim (x',y')$ if and only if $y = y'$. Now prove that the quotientspace $X/\sim$ with ...
2
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1answer
24 views

X={1,2,3}. Give a list of topologies on X such that every topology on X is homeomorphic to exactly one on your list.

I'm teaching my self topology with the aid of a book. I'm trying to do the following problem: Let X={1,2,3}. Give a list of topologies on X such that every topology on X is homeomorphic to ...
2
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1answer
23 views

Defining compact sets with closed covers

This question is a continuation of this. My book says that a metric space is compact if and only if: $$M=\cup A_{\lambda}\implies M = A_{\lambda1}\cup\cdots\cup A_{\lambda_n}$$ where each $A_{\...
1
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1answer
23 views

Show excluded point topology is a topology

I'm teaching my self topology with the aid of a book. I'm trying to prove the following is a topology: Let X be an infinite set, and $p$ be an arbitrary point in $X$. Show that $\mathscr{T}_4=\{U \...
3
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1answer
36 views

Connectedness of Product Topology

I am working on a pretty straight forward proof, I am trying to show that when I have a family of topological spaces $ (X_i, T_i )_{i \in I}$ where all $(X_i, T_i )$ are connected that the product ...
1
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2answers
23 views

Show particular point topology, is a topology

I'm teaching my self topology with the aid of a book. I'm trying to prove the following is a topology: Let X be an infinite set, and $p$ be an arbitrary point in $X$. Show that $\mathscr{T}_3=\{U \...
0
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1answer
26 views

Confusion about difference between Normal and Perfectly Normal

I am working with the following definitions: A topological space is normal if and only if every pair of disjoint, nonempty closed sets can be separated by a continuous function. A ...
0
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0answers
21 views

Prove that the cantor space is totally disconnected.

Prove that the cantor space is totally disconnected. Let $(G,T)$ be the Cantor space and let $\prod_{i=1}^{\infty}(A_i,T_i)$ be homeomorphic to the Cantor space where $(A_i,T_i) = (\{0,2\}, T_{...
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4answers
41 views

Correctness of proof that every neighborhood is an open set.

Rudin makes the following definitions: (a) A neighborhood of p is a set $N_r(p)$ consisting of all $q$ such that $d(p, q) < r$, for some $r > 0$. (b) $E$ is open if every point of $E$ is an ...
0
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1answer
38 views

Baire space using extended metric?

Consider the set $C^1(\mathbb{R},\mathbb{R})$ of continuously differentiable functions on $\mathbb{R}$, endowed with the extended $C^1$ norm $\|f\|_{C^1} = \sup_{x\in \mathbb{R}} |f(x)| + \sup_{x\in \...
2
votes
2answers
57 views

Consequence of Riesz Representation Theorem from Rudin RCA

It's Riesz Representation Theorem from Rudin's book. In the following chapter I met the following example: It's obvious that $\sigma$-compact set has the $\sigma$-finite measure. But how to prove ...
2
votes
1answer
37 views

Show “countable complement topology” is a topology

I'm teaching my self topology with the aid of a book. I'm trying to prove the following is a topology: Let X be an infinite set. Show that $\mathscr{T}_2=\{U \subseteq X : U = \emptyset $ or $ X\...
0
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2answers
16 views

Claim $(\mathbb{N}, \leq)$ is a discrete space, but is $(-\infty, b)$ a subbasic element?

Let $\mathbb{N}$ denote the set of natural numbers, then a subbasis on $\mathbb{N}$ is $$S = \{(-\infty, b), b \in \mathbb{N}\} \cup \{(a,\infty), a \in \mathbb{N}\}$$ Let $\leq$ be the relation on ...
0
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1answer
27 views

compact metric space definition by closed covers

My book says the following: A metric space is compact iff: $$M=\cup A_{\lambda}\implies M = A_{\lambda1}\cup\cdots\cup A_{\lambda_n}$$ where each $A_{\lambda}$ is open. Then, it says that if $A_\...
3
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1answer
36 views

Compact subset in an infinite product space (Theorem 5.16 from General Topology by Kelley)

I found the following theorem on page 145 of Kelley's General Topology: If an infinite number of the coordinate spaces are non-compact, then each compact subset of the product is nowhere dense. ...
0
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1answer
41 views

CW complex= cell complex?; compact metrizable spaces

I have a question about finite cell complexes and compact metrizable spaces. In a paper I read the statement: Let $X$ be a compact metrizable space. Then $X$ is a countable inverse limit $\varprojlim\...
0
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1answer
46 views

Is it my error or the term “normal” has multiple meanings?

I use a definition of normal quasi-uniform spaces from this article. Now I have proved (I do not present the proof because it uses "funcoids" which can be read about only in my manuscripts.) that ...
1
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1answer
34 views

Given a topological space $X$, is it possible to define a topology on $A^X$

Assume that $X$ is a topological space and $A^X = \{f\mid f:X\rightarrow A\}$ consists of all functions from $X$ to an arbitrary set $A$. Is it possible to define a natural (inherited from $X$ i.e. it ...
0
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1answer
42 views

Version of Invariance of Domain for n-manifolds

I am working on the following exercise from Lawson's Topology: A Geometric Approach: Apply Invariance of Domain (If $U$ is an open subset of $\mathbb{R}^n$ and $f:U\rightarrow\mathbb{R}^n$ is $1$-$1$...
0
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1answer
29 views

Closure of sequence having finite non zero terms

Consider $\Bbb{R}^{\omega}$ in product topology and uniform topology where the uniform topology is generated by the uniform metric $$\overline{\rho}((x_n), (y_n))= \sup \{ |x_n-y_n|\}.$$ Let $S$ be ...
0
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1answer
29 views

Every completely regular space is regular

The definitions I'm working with: $(X,\mathcal{T})$ is said to be completely regular if for every $x \in X$ and every closed set $C \subseteq X$ not containing x can be separated by a continuous ...
0
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1answer
51 views

Find the largest possible radius for $S=[1,4) \cup (4,9]$

Find max $\{\epsilon : N(x;\epsilon) \subseteq S\}$, the largest $\epsilon$ such that the neighborhood centered at $x$ of radius $\epsilon$ is contained in $S$. That is, state the largest possible ...
0
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0answers
50 views

Is the cohomology ring of a CW complex computable?

There is a well-developed technology for computing the cohomology groups of a CW complex, cellular cohomology. It reduces the problem of computing cohomology to the two simpler problems of (1) ...
1
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1answer
35 views

Is it sufficient to prove that a function is an open map by looking at the basis element?

I am trying to prove that the projection map $\pi_X:(X, T)\times (Y,J) \to X$ is an open map But I don't know if I can use the basis element directly, so my proof is quite round about and lengthy ...
5
votes
1answer
91 views

Ref. Requst: Space of bounded Lipschitz functions is separable if the domain is separable.

I have been scouring the internet for answers for some time and would therefore appreciate a reference or a proof since i'm not able to produce one myself. Let $(\mathcal{X},d)$ be a metric space, ...
1
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1answer
15 views

Show that there exists at most one extension of $f$ whose co-domain is a Hausdorff space [duplicate]

I want to show the following Suppose $A \subset X, f: A \to Y$ is continuous, $Y$ is Hausdorff. Show that there is at most one continuous extension $g: \overline A \to Y$ I feel like I am ...
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2answers
44 views

How to see that $f(t) = (t, 2t, 3t, \ldots)$ continuous in the product topology

I am trying to check whether $f: \mathbb{R} \to \mathbb{R}^\omega$ $f(t) = (t, 2t, 3t, \ldots)$ is continuous or not in the product and box topology. But I have a feeling I don't have the ...
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1answer
40 views

Non-separable metrisable space [closed]

I'd like to see counter-examples to the following (false) statements: 1) A metrisable topological space is separable. 2) All arc-connected topological spaces are contractible. 3) If $X$ is compact ...
3
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3answers
68 views

Topology: is it ever good to write $x \in U \in \mathfrak{T}$

Sometimes I come across a sentence in my topology book that says, let $U$ be an open set that contains $x$ I can't help but write it down as: Let $$x \in U \in \mathfrak{T}$$ Is it good ...
0
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2answers
66 views

Does $(0,1)$ embed in $[0,1]$ through inclusion or identity?

Given two spaces $X,Y$, we say that $X$ is embedded in $Y$ if there is a homeomorphism $f: X \to A \subseteq Y$ My question is let $X = (0,1)$, and $Y = [0,1]$, does $X$ embed in $Y$ by inclusion map ...
0
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0answers
23 views

Invariance of Linking numbers and critical values

So, I am trying to show that for a map $f: S^{2p-1} \rightarrow S^p$ , the linking number $l(f^{-1}(y),f^{-1}(z))$ of two framed submanifolds associated with regular values $y,z$ of $f$, defined as ...
1
vote
1answer
36 views

Arithmetic progressions form infinite basis on $\mathbb{Z}$

Let $B(a,b) = \{ax+b: a,b \in \mathbb{Z}, a \neq 0, x \in \mathbb{Z}\}$ be a so called arithmetic progression I am required to show that that $\mathcal{B} = \{B(a,b) | a,b \in \mathbb{Z}\}$ is a ...
7
votes
2answers
56 views

Prove that $\overline{f(A)}\subseteq f(\overline{A})$ where $f: X \rightarrow Y$ is continuous, $X$ is compact and $A \subseteq X$

Suppose that $X$ and $Y$ are topological spaces, $f: X \rightarrow Y$ is a continuous map and $A \subseteq X$. It's not very hard to prove that $f(\overline{A})\subseteq \overline{f(A)}$, where $\...
0
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4answers
67 views

Are compact sets on $\mathbb R^n$ always connected?

I am unsure if compact sets on $\mathbb R^n$ are always connected. Can someone explain it to me?
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3answers
38 views

Compact set and a sequence of closed sets, the intersection of all of them is empty

I'm having a hard time to prove this. The problem is: Let $V \subset \mathbb{R}^d $, be a nonempty compact set and $(A_n)_{n\in\mathbb{N}}$ a sequence of closed nonempty sets in $\mathbb{R}^d$ with ...
-3
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0answers
25 views

infinite product of topological spaces open sets [closed]

I am refering to the definition of infinite product of topological spaces that states that the most of the open sets used for the topology will be whole spaces except a finite number of spaces from ...
0
votes
1answer
38 views

Assistance with finding the accumulation points for $(3,6) \cup (6,9]$

I'm having trouble digesting the definition of an accumulation point(s). Can you help me to understand it given the following: $(3,6) \cup (6,9]$ I know this produces the interior set $(3;9]\...