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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2answers
28 views

If $\forall V\subseteq X$ where $x\in \overline V; f(x) \in \overline{f(V)}$, then $f$ is continous in $x$

Let $f:(X,\tau_X)\to (Y,\tau_Y)$ Prove: If $\forall V\subseteq X$ where $x\in \overline V; f(x) \in \overline{f(V)}$, then $f$ is continous in $x$. Could someone verify the following proof? ...
1
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1answer
31 views

Closeness of infinite union of closed sets

Is the set $\bigcup_{x \geq 0} \left\{\frac{1}{x+1} \right\} $ closed? For all $x \geq 0$, the set $\left\{\frac{1}{x+1}\right\}$ is a single point, therefore it is closed. But I am not sure about ...
5
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2answers
96 views

Is the closure of $\mathbb Q \times \mathbb Q$ equal to $\mathbb R \times \mathbb R$?

I know the closure of $\mathbb Q$ is $\mathbb R$, but does this imply that the closure of $\mathbb Q \times \mathbb Q$ equal to $\mathbb R \times \mathbb R$?
6
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3answers
65 views

Prob 12, Sec 26 in Munkres' TOPOLOGY, 2nd ed: How to show that the domain of a perfect map is compact if its range is compact?

Let $X$ and $Y$ be topological spaces such that $Y$ is compact, and let $f \colon X \to Y$ be a closed, surjective, and continuous map such that, for each $y \in Y$, the inverse image $f^{-1} ( \ \{y ...
0
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0answers
37 views

Cone of projective space?

Is the cone of $\mathbb{C}\mathbb{P}^2$ a familiar topological space? What about $\mathbb{C}\mathbb{P}^3$? I'm having a lot of trouble visualizing it. I just learned the notion of the cone of a ...
2
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0answers
68 views

Which spaces admit bump functions?

Let me first fix terminology. Let $X$ be a topological space and $A\subseteq B\subseteq X$ be subsets. Let's say X admits a bump function relative to $A\subseteq B$ if there's a continuous function ...
6
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1answer
81 views

What is the class of topological spaces $X$ such that the functors $\times X:\mathbf{Top}\to\mathbf{Top}$ have right adjoints?

For any topological space $X$, define a functor $\times X:\mathbf{Top}\to\mathbf{Top}$ by $Y\mapsto Y\times X$ (and acting on the hom-sets in the natural way). I know that if $X$ is locally compact, ...
0
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1answer
21 views

Boundary of union of disjoint open sets.

Let $X$ be a topological space. For any subsets $A,B \subseteq X,$ $\partial (A \cup B) \subseteq \partial A \cup \partial B.$ In general, they are not equal. When $A$ and $B$ are disjoint and open, ...
1
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2answers
42 views

Prove equivalence between $X$ Hausdorff and $X$ finite with discrete topology

We have a Noetherian topological space $X$. Show that the following are equivalent: $X$ is a Hausdorff space $X$ is finite and has discrete topology So far I've only got this: If $X$ has discrete ...
1
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0answers
24 views

whether any shape can be placed on a tiled surface?

After read "Prove that any shape 1 unit area can be placed on a tiled surface",I think on a surface of equal square tiles where each tile side is 1 unit long,the shape ,less than some constant C>1 ...
1
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1answer
31 views

Normed space of bounded functions $f:\mathbb{N}\to\mathbb{N}$

Let $X = \{f:\mathbb{N}\to\mathbb{N}: \exists M\in\mathbb{N} \forall n\in\mathbb{N} f(n) \leq M\}$. Define a norm on $X$ by defining for $f\in X$: $$||f|| = \sum_{n=1}^\infty \frac{f(n)}{2^n}.$$ Is ...
-1
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0answers
35 views

Proofs involving 3 quantifiers: A(3,3)=6 cases [closed]

I knows how to prove statement involving 1 or 2 quantifiers. So there are 6 combinations of 3 universal quantifiers ("for all" and "there exist") with an extra implication that makes a quantified ...
4
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2answers
317 views

Modern mathematics for dummies

I have poor university mathematical education, but Math fascinates me, so I decided to educate myself for a bit. I know there is a dozen modern mathematical fields I know nearly nothing about like ...
3
votes
1answer
58 views

A $G_δ$ subset of $2^ω$ that is homeomorphic to $ω^ω$

How do I show that there is a $G_δ$ subset of the Cantor space $2^ω$ that is homeomorphic to the Baire space $ω^ω$? I've been given the hint to consider $G = \{x ∈ 2^ω : x\text{ is not eventually ...
3
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1answer
33 views

Question about the Baire space, $\sigma$-algebra and $\sigma$-ideal.

Let $\text{BP}(X)$ denote $\sigma$-algebra of subsets of $X$ with the Baire Property BP and $\text{MGR}(X)$ denote the $\sigma$-ideal of meager sets in $X$. Assume $X$ is second countable Baire ...
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1answer
162 views

What can we say about closed sets in the Baire space that are neither open nor compact?

I'm trying to figure out what closed subsets in $\omega^{\omega}$ equipped with product topology should look like. It seems to me it's relatively easy to have an idea about compact closed subsets and ...
1
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1answer
35 views

Closed set in Baire space

I am reading a book on Banach spaces. It introduces the Baire space $\mathcal{N}=\mathbb{N}^\mathbb{N}$ as the product of infinitely many copies of $\mathbb{N}$ with the discrete topology. We have ...
1
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1answer
59 views

Question about of Baire property and Baire space

In reading Kechris book. Please, I would like help with this proposition. For convencion we put for $A \subseteq X$, $$\sim A=X\setminus A$$ If $A$ is comeager in $U$, we say that $U$ forces $A$, ...
0
votes
1answer
26 views

Connection between one-point compactification of $\mathbb{R}$ and $S^1$

Definition I got: A one-point compactification of $X$ is $\hat{X}=(X\cup\{\infty\},\tau).$ The new topology $\tau$ is generated by open subsets of $X$, and $U\cup\{\infty\}$ where $U=K^C$ where $K$ ...
4
votes
2answers
90 views

Is $\Bbb R$ the soberification of $\mathbb{Q}$?

I'm a beginner. I read about soberification of topological space and thought that if I soberificate $\mathbb{Q}$, for any $x \in \mathbb{R}$, the neighbourhood filter of $x$ in $\mathbb{Q}$ ...
2
votes
1answer
55 views

Characterization of closed map by sequences/nets

I'm interested in characterizing closed maps in terms of nets. Since a map is closed iff $\overline{f(V)} \subseteq f(\overline{V})$ for all subsets $V$, I believe one possible such characterization ...
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1answer
31 views

The intersection of a locally finite family of open sets is a closed set?

Let $\{ U_i : i \in I \}$ be a locally finite family of open sets, if $I$ is infinite index, and there exist infinite different $U_i$, must $\bigcap_{i \in I} U_i $ be a closed set? Or, further more, ...
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1answer
31 views

Intuition for order type

Here is a definition from Munkres' book "Topology": For any two sets $A$ and $B$ with order relations $<_A$ and $<_B$, respectively, we say that $A$ and $B$ have the same order type if there ...
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0answers
18 views

Martin's axiom solves Ponomarev's problem

I'm looking for the following paper: Juhász, I. Martin's axiom solves Ponomarev's problem. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 18 1970 71–74. Thanks for your help.
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0answers
38 views

Prove that a finite union of closed sets is also closed (using limit points)

Let $F_i$ be a family of closed sets, then we know that $\bigcup_{i=1}^nF_i$ is closed. Proving that statement is equivalent to proving: If $p$ is a limit point of $\bigcup_{i=1}^nF_i$ then ...
2
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0answers
62 views

Union of infinite broom and topologist's sine, connectednes, locally connectednes properties…

I'd like to know if my answer of the following exercise is correct. I really appreciate any suggestion you can provide to improve my argument or corrections in case I made a mistake :) Let ...
1
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0answers
16 views

Continuity of Component Function

Let $f:Z\times X \to Y$ be given such that $f$ is continuous. I'm trying to prove that $f(z, -)$ is continuous for a fixed $z\in Z$. I would appreciate if someone could tell me if the proof that ...
0
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0answers
16 views

There is an open set containing only some distinct prime ideals of $\operatorname{spec}(R)$

I want to show that given a ring in which every prime ideal is a maximal ideal and provided some distinct (!) prime ideals $p_1, \dots, p_n, \hat{p}_1, \dots, \hat{p}_m \in \operatorname{spec}(R)$, I ...
3
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2answers
71 views

Theorem 7.2 in General Topology by S. Willard

Theorem 7.2 If $X$ and $Y$ are topological spaces and $f:X \to Y$ , then the following are all equivalent :- I) $f$ is continuous. II) for each E $\subset X$ , $f(\bar E) \subset ...
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0answers
39 views

Prove that $f|A_i$ continuous for closed sets $A_i$ if and only if $f$ is continuous [duplicate]

Let $(X,\mathcal{O})$ be a topological space, $(A_i)_{i\in I}$ be closed sets and assume $X = \bigcup_{i\in I} A_i$, For all $x \in X$ there exists an open set $O$ with $x \in O$, so that $O ...
3
votes
0answers
44 views

Composition of functions in Munkres' Topology

In Munkres' book "Topology", he writes that: Given functions $f:A\to B$ and $g:B \to C$, .... the composition $g \circ f $ is defined only when the range of $f$ equals the domain of $g$. But, ...
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0answers
35 views

Topologically equivalent metrics, using different definitions.

I´ve been dealing with topologically equivalent metrics for a while, using the usual definition, that $d$ and $d'$ are topologically equivalent iff they have the same open sets. However, there is ...
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1answer
27 views

it is possible to define a topology on particular vector space

If i take $V$ a finte dimensional vector space on the real number (or complex number). Setting $n=dim_{\mathbb{R}}(V)$, i know that there is a isomorphism of vector spaces so $V \simeq \mathbb{R}^n$. ...
0
votes
1answer
28 views

Proof of the Harnack inequality

Let $\Omega\subseteq\mathbb{R}^n$ be a domain, $\Omega'\subset\subset\Omega$ be a domain and $u\in C^0(\overline{\Omega})$. Suppose we know $$\sup_{\Omega'}u\le 3^n\inf_{\Omega'}u\tag{1}$$ if ...
-1
votes
1answer
60 views

$A$ and $B$ compact in a Hausdorff space implies $A\cap B$ is compact [closed]

Prove that if $A$ and $B$ are compact subset of a Hausdorff space $X$, then $A$$\cap$$B$ is compact.
0
votes
1answer
64 views

Prerequisites for learning general topology

I want to learn general topology in order to apply it in electromagnetism. I am an undergraduate student and I have a background in linear algebra (not at an advanced level), linear differential ...
14
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7answers
3k views

Choosing a text for a First Course in Topology

Which is a better textbook - Dugundji or Munkres? I'm concerned with clarity of exposition and explanation of motivation, etc.
0
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1answer
23 views

nonempty disjoint closed subsets of $\mathbb{R}^{n}$ [closed]

Let $A$ and $B$ be two nonempty disjoint closed subsets of $\mathbb{R}^{n}$. For $x\in{\mathbb{R}^{n}}$ let $f(x)={d(x,A)/{(d(x,A)+d(x,B))}}$ Show that $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ is ...
0
votes
0answers
34 views

Two problems about closed set in $\Bbb R^2$ [closed]

Let $F$ be a closed set in $\Bbb R^2$, $F\neq \varnothing,\Bbb R^2$, and $F^\circ\neq \varnothing$。 Problems: Is $F^{\circ-}-F^\circ$ a perfect set in $\Bbb R^2$? Let $L$ be the disjoint union of ...
2
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1answer
24 views

What does compactness of $\mathbb R$ under one of these topologies imply about compactness under the other?

Let $\tau ,\tau_1$ be two topologies on the set $\mathbb R$ .Suppose $\tau \subset \tau_1$ .What does compactness of $\mathbb R$ under one of these topologies imply about compactness under the other? ...
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0answers
75 views

The skeleton of Eulerian polyhedra

There is (at least) two kind of validity domain of Euler's $v−e+f=2$ polyhedron formula. One is the "Eulerian" polyhedra, i.e. simply connected polyhedra with simply connected faces (see here). The ...
11
votes
3answers
120 views

Can we fit uncountably many nonempty open sets in $\mathbb{R}^n$ such that each point is contained in at most finitely many of them?

This simple question came to my mind the other day: Question: Can we fit uncountably many nonempty open sets in $\mathbb{R}^n$ such that each point of $\mathbb{R}^n$ is contained in at most ...
3
votes
2answers
100 views

For an open subset $U\subseteq X\times Y$, is the section $U_x$ open in $Y$?

Let $X,Y$ be topological spaces and let $U$ an open subset of $X\times Y$. For $x\in X$, let $$U_x=\{y\in Y: (x,y)\in U\}$$ Is it $U_x$ an open subset of $Y$? Thanks.
1
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1answer
53 views

When a sigma-finite space is a sigma-compact space?

$X$ is a topological space, $m$ is a $\sigma-$finite measure on $B(X)$, and what condition can make $X$ be a $\sigma-$compact space? This question is from topological groups (for me). Locally compact ...
3
votes
1answer
35 views

Let $f: (X, d) \to (Y, d')$. Prove that the following are equivalent:

Let $f: (X, d) \to (Y, d')$. Prove that the following are equivalent: a) $f$ is uniformly continuous in $X$. b)For every pair of sequences $(x_n), (y_n) \subseteq X$ such that $ d(x_n, ...
3
votes
1answer
35 views

Show that $A=\bigcap G_{A}$

Given a metric space $(X,d)$ and $A\subset X$, let $G_{A}$ be the set which consists of all the open sets that contain $A$. Show that $A=\bigcap_{B \in G_{A}}B$ It is obvious that $A \subset ...
2
votes
2answers
32 views

Inverse mapping on a set $U_1\times U_2$, wrong intuition?

Let $f(x) = (f_1(x),f_2(x))$ where $f: X\to Y_1\times Y_2$. And $f_1:X\to Y_1, f_2: X\to Y_2$ where $X,Y_1,Y_2$ are topological spaces. I want to prove some continuity properties, but my ...
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votes
1answer
65 views

Is Cantor set closed? [closed]

https://www.youtube.com/watch?v=dazO9UoKmyA This professor said Cantor set is closed because it's FINITE union of closed intervals at 14.00. But isn't it a wrong statement since Cantor set is ...
1
vote
1answer
15 views

Covers of X in the Stone-Cech compacticatoin

Suppose that $X$ is a Hausdorff completely regular space. $X$ embeds homeomorphically into its Stone-Cech compactifaction $\beta{X}$ and is dense in $\beta{X}$; we can identify $X \subset ...
0
votes
2answers
18 views

countable product of totally bounded space is totally bounded

Let $ \{ X_i \}_ { i \in \mathbb{N} }$ be a countable collection of metric spaces $(X_i, d_i)$. The product topology on product space $X=\prod_{i=1}^{\infty} X_i$ is equivalent to the metric topology ...