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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2
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1answer
237 views

Fixed point of a continuous map

Take $X = [0,1]$, and a continuous map $f:X \rightarrow X$. Then there exist a point $x \in X$ s.t. $f(x) = x$. We may take $X = (0,1)$ or $X = (0,1]$. Shall we get such fixed points in latter cases? ...
0
votes
1answer
97 views

KC-spaces and US-spaces.

A topological space is called a US-space provided that each convergent sequence has a unique limit. A topological space is called a KC-space provided that every compact subset is closed. So ...
1
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1answer
76 views

topological KC - space

A topological space X is KC – space if every compact subsets are closed. question: Does a KC - space contains a minimal KC topology?
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1answer
69 views

minimal KC and (strongly) KC

If P is a topological property, then a space (X, τ) is said to be minimal P (respectively, maximal) if (X,τ) has property P but no topology on X which is strictly smaller (respectively, strictly ...
0
votes
1answer
92 views

topological space

Let $( X,\tau )$ be a $T_1$ topological space. Let $D = \{ d_n : n \in \omega \}$ be a countably infinite closed discrete subspace of $X$. Fix $P \in X$ and let $F \in \beta\omega- \omega$ be an ...
5
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2answers
130 views

Orthogonal chords of compact sets

For any compact set on a plane say C does there always exist a chord in C such that its end points are orthogonal to the boundary of C (assumed smooth)
0
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1answer
254 views

Properties for interior and closure in metric space.

I found the some following properties for general topology and prove these. But, I want to verify that the proofs are really true. Let $(X,d)$ be metric space. Let $A$ be any subset of $X$. Define ...
2
votes
1answer
75 views

How to prove this version of Urysohn's Lema using the usual version

I'm trying to prove the Urysohn's lemma that is presented in Rudin's Real and Complex Analysis using the usual version of the Urysohn's Lemma (with normal space ...). Here is the Rudin's Version: ...
0
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1answer
134 views

Is a single point boundaryless?

I am trying to understand Preimage orientation. So I got this question: Definition. The boundary of $X$, consists of those points that belong to the image of the boundary of $\mathbf{H}^k$, the ...
1
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3answers
3k views

Proving Every open set in $\Bbb R$ is a countable union of open intervals. [duplicate]

This question is from William R. Wade's Introduction to Analysis book: Prove that every open set in $\Bbb R$ is a countable union of open intervals. I have no ideas honestly. Thank you.
4
votes
1answer
583 views

Does a compact connected complete linear order have the fixed point property?

Would the same arguments used for showing $[0,1]$ has the fixed point property hold in this general case? What could go wrong? EDIT: The fixed point property can be interpreted in two ways that I ...
4
votes
1answer
80 views

Prove that $H$ is compact $\iff$ every cover $\{E_{\alpha}\}_{\alpha \in A}$ has a finite subcovering.

Let $H \subseteq \Bbb R^n$. Prove that $H$ is compact $\iff$ every cover $\{E_{\alpha}\}_{\alpha \in A}$ where $E_{\alpha}$'s are relatively open in $H$ has a finite subcovering. $\bf{Solution \ ...
3
votes
2answers
74 views

Prove that $H$ is a finite set.

Let $H$ be compact in $\Bbb R^n$ Also assume that for every $x\in H$ there is an $r=r(x)$ such that $B_r(x)\cap H=\{x\} $ Prove that $H$ is a finite set. Solution: Since $H$ is compact, ...
1
vote
1answer
66 views

Show that $Y$ is locally compact.

Recall that a space $X$ is locally compact if for any point $x$ in $X$, and any neighborhood $U$ of $x$ in $X$,there is a neighborhood $V$ of $x$ in $X$,and a compact subspace $C$ of $X$ such that $x ...
4
votes
2answers
103 views

Prove that f is constant on $K$ that is, if $a \in K$ then $f(x)=f(a) \ \ \forall x\in K$

Suppose that $f: \Bbb R^n \to \Bbb R^m$ and that $a\in K$, where $K$ is a compact connected subset of $\Bbb R^n$ suppose for each $x\in$ $K$, $\exists$ $\delta_x >0$ such that $f(x)=f(y)$ ...
2
votes
3answers
154 views

Prove that $a$ is a cluster point of $E$ $\iff$ for each $r>0$, $E\cap B_r(a)$ \ $\{a\}$ is nonempty.

Question: Let $E$ be a subset of $\Bbb R^n$ Prove that $a$ is a cluster point of $E$ $\iff$ for each $r>0$, $E\cap B_r(a)$ \ $\{a\}$ is nonempty. definiton: A point $a \in \Bbb R^n$ is ...
5
votes
1answer
256 views

Quantifiers as Adjoints in Generalized Logics

It is a well known fact that the classical universal and existential quantifiers can be seen as adjoints in certain categories. In the continuous model theory of metric structures (see ...
4
votes
2answers
290 views

Prove that every closed ball in $\Bbb R^n$ is sequentially compact.

Question: Prove that every closed ball in $\Bbb R^n$ is sequentially compact. A subset $E$ of $\Bbb R^n$ is said to be squentially compact $\iff$ every sequence $x_k\in E$ has convergent ...
0
votes
2answers
119 views

Structure of topological spaces in terms of sequences

I been reading for several hours and not yet found a question put in this way. Given any topological space: Does every sequance in $X$ determine a countable subset of $X$? Do the sets that belong ...
2
votes
1answer
78 views

How to prove that initial arrows in Haus coincide with topological embeddings?

In Joy of Cats it is stated that in category $\textbf{Haus}$ initial arrows coincide with topological embeddings (pg 135). This can be proved by showing that initial arrows in $\textbf{Haus}$ are ...
4
votes
1answer
206 views

Prove that a proper subset $E$ of $\Bbb R^n$ is connected $\iff$ it contains exactly two relatively clopen sets.

Prove that a proper subset $E$ of $\Bbb R^n$ is connected $\iff$ it contains exactly two relatively clopen sets. I researched the meaning of "clopen set". And I reached the result that so as to ...
5
votes
2answers
97 views

domain of initial $f : X \rightarrow Y$ in Haus equipped with coarsest topology?

If $f:X\rightarrow Y$ is initial in category Top then it is easy to proof that (!) the topology on $X$ is the set of preimages of open sets in $Y$. Just construct topology $Z$ having the same ...
1
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1answer
43 views

Question on Computation of Integral of a Form

Again: I'm trying to understand the result of a certain integral of a form in a paper I'm reading (for which I do not, unfortunately, have a link): We start with a surface S that is oriented, ...
2
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2answers
84 views

Compact and countability axioms!

I wondering which countability axioms compact imply in arbitrary topological spaces. I'm using Greene/Gamelin 2nd ed. And they list separable, 2nd-coutable, first-countable and Lindelöf. Clearly ...
3
votes
1answer
57 views

On the proof of deformation lemma “boundedness”

Book- Evans partial differential equation. In the proof of deformation lemma how to say that $V(u)=-g(u)h(\lVert I'(u)\rVert)I'(u)$ is bounded. And how to say that the mapping $u \to ...
1
vote
1answer
150 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 ...
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3answers
89 views

Checking for uniform convergence $f_{n}(x)$ $=$ $n^{2}x\ (1-x)^{n}$

Let $f_{n}(x)$ $=$ $n^{2}x\ (1-x)^{n}$ with $0\le x\le 1$, where $f_{n}$ converges pointwise to the zero function $f$. How do I check for uniform convergence? Can someone provide me with some hints? ...
2
votes
4answers
100 views

Closure of a certain subset in a compact topological group

Suppose that $G$ is a compact Hausdorff topological group and that $g\in G$. Consider the set $A=\{g^n : n=0,1,2,\ldots\}$ and let $\bar{A}$ denote the closure of $A$ in $G$. Is it true that ...
3
votes
1answer
53 views

Why do the author added the extra condition that $X$ needs to be $T_1?$

In my text it's written that, But I get to prove the result underlined red simply for a first countable space as: (N.B. by limit point the author wanted to mean the adherent point) ...
2
votes
1answer
55 views

Is there any way to prove it directly?

I'm trying to prove the following result: In a first countable $T_1$ space $X$ for $E\subset X,~x\in X$ is an adherent point of $E\iff~\exists~(x_n)_n\in E$ such that $x_n\to x.$ When I'm ...
0
votes
1answer
74 views

How to show this line is tangent to $f$ at point $a$?

Let $f:I\to\mathbb{R}^n$ be a differentiable function, with $f'(a)\neq 0$ for some $a$ in the interval $I\subset\mathbb{R}$. If there exists a line $L\subset\mathbb{R}^n$ and a sequence $(x_k)$ in ...
8
votes
2answers
246 views

Examples of closed subspaces of Baire spaces that fail to be Baire?

I am looking for some nice examples of Baire spaces containing closed subspaces that fail to be Baire. Clearly, $X$ should not satisfy either one of the standard hypotheses for the Baire category ...
2
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1answer
398 views

Non-empty intersection of open balls in $R^n$ contain open balls

I want to prove that if the intersection of two open balls about the points $x, y$ (resp.) is non-empty, then there exists a third ball centered at some point $z\in B_{\epsilon 1}(x)\cap B_{\epsilon ...
3
votes
1answer
139 views

Quaternionic general linear group is open

Is there an elegant proof of the following fact: "The quaternionic general linear group $GL(n, \mathbb{H})$ is open in $M_n(\mathbb{H})$", where $M_n(\mathbb{H})$ is the set of all $n \times n$ square ...
1
vote
1answer
69 views

Lifting homeomorphisms covering

Hello I had a question regarding a lemma from the paper: http://www.math.columbia.edu/~jb/bir-hilden-annals.pdf I don't understand the proof of Lemma 5.1. Notation: $T_{0,0}$ is the 2-sphere, ...
0
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1answer
84 views

Kind of a silly question, but need confirmation regarding the closed unit interval $[0,1]$

I know that $[0,1]$ is a closed subset of $\mathbb{R}$, since its complement $(-\infty,1) \cup (1, \infty)$ is open in $\mathbb{R}$. Clearly, $(0,1)$ is an open subset of $\mathbb{R}$, but is it an ...
2
votes
2answers
144 views

The Stone-Čech compactification of a space by the maximal ideals of the ring of bounded continuous functions from the space to $\mathbb{R}$

There is a claim that for any completely regular space, the maximal ideals of the ring of bounded continuous functions from $X$ to $\mathbb{R}$ forms the Stone-Čech compactification of $X$. How is the ...
6
votes
4answers
515 views

Why the axioms for a topological space are those axioms?

This question might have even been asked here before, I don't really know, so sorry if it's duplicate. I've started to study topological spaces and I've found the axioms for such spaces kind of hard ...
0
votes
1answer
212 views

Let $A,B$ be nonempty subsets of a topological space $X$. Prove that $A\cup B$ is disconnected if $(\bar{A}\cap B)\cup(A\cap\bar{B})=\emptyset$.

I'm reading Intro to Topology by Mendelson. The problem statement is, Let $A,B$ be nonempty subsets of a topological space $X$. Prove that $A\cup B$ is disconnected if $(\bar{A}\cap ...
2
votes
2answers
294 views

Let $A,B\subset X$, $X$ a topological space. If $A$ is connected, $B$ open and closed, and $A\cap B\neq\emptyset$ then $A\subset B$.

I'm studying Intro to Topology by Mendelson. The problem statement is, Let $A,B\subset X$, $X$ a topological space. If $A$ is connected, $B$ open and closed, and $A\cap B\neq\emptyset$ then ...
3
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1answer
212 views

On the real line, prove that the set of nonzero real numbers is not a connected set.

I'm studying Intro to Topology by Mendelson. The problem is stated in the title. My proof is, Let $A=\mathbb{R}-\{0\}$. Then $C(A)=\{0\}$. Moreover, let $P=(0,\infty)$ and $Q=(-\infty,0)$. Then ...
0
votes
1answer
112 views

Homeomorphism under subspace topology in Hausdorff space

Let $Y$ be a Hausdorff space, and $U,V \subset Y$ are homeomorphic under subspace topology. Does this imply if $U$ is open(or closed) then $V$ is open(or closed) under original topology? I can't ...
2
votes
1answer
458 views

Finding limit points of subsets of the cofinite topology on $\mathbb{Z}$

Is my reasoning correct? Problem: Let $(Z,\tau)$ be the cofinite topology on $Z$. Find the limit points of the sets: $A = \{1,2,\dots,10\}$ $E$, the even integers My solution: $A$ is closed ...
16
votes
1answer
408 views

How do Slinkies become tangled?

The following image describes the problem better than I can: As you know, sometimes Slinkies can twist such that the direction of the coil can be reversed. However, though reversed, the coil still ...
2
votes
1answer
45 views

$C$-embedding in uniform spaces

Every Hausdorff uniform space $X$ has a Hausdorff completion $C_X$. Is it true that $X$ is $C$-embedded in $C_X$? How about the completion with respect to its finest uniformity $\mu_X$?
6
votes
5answers
246 views

This set of matrices is open

I'm trying to prove that the set of the matrices whose eigenvalues have non-zero real part is an open subset of $M^n$, the set of square matrices with order $n$ which is identify with $\mathbb ...
1
vote
1answer
64 views

Weak star limit in $L^{\infty}(0,T;L^2(\Omega))$

Suppose that you know that $v_N$ is such that : $\forall N\in \mathbb{N}$, $v_N \in \mathcal{C}^0(0,T;L^2(\Omega))$ and $\partial_t v_N \in L^{\infty}(0,T;L^2(\Omega))$ ($\Omega$ an open bounded ...
5
votes
2answers
103 views

Pre-images of closed sets are open

Let $X$ and $Y$ be two topological spaces and let $f$ be such a map that $f^{-1}(A)$ is open in $X$ for any closed $A$. Note that if $X\stackrel{f}{\longrightarrow}Y\stackrel{g}\longrightarrow Z$ are ...
13
votes
3answers
297 views

Is there a suitable definition in categories for a closed continuous function in topology?

Working in the category of topological spaces is it possible to give a 'categorical' definition for 'a closed continuous function'? I mean something like: 'a closed continuous function' is an arrow in ...
1
vote
1answer
52 views

Question on boundedness

Page number 479 in partial differential equation by Evans book how to say that the derivative of I is bounded on bounded sets