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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5
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0answers
82 views

Topological Space in which every compact subset is metrizable

Is there an (more or less) established name for the class of topological spaces in which every compact subset is metrizable? This is true for example in (LF)-spaces (inductive limits of ...
2
votes
2answers
70 views

n×n matrices A with complex enteries

Let U be set of all n×n matrices A with complex enteries s.t. A is unitary. then U as a topological subspace of $\mathbb{C^{n^{2}}} $ is compact but not connected. connected but not compact. ...
2
votes
2answers
72 views

Is it necessarily a closed subset?

Let $ f:X \rightarrow Y$ be a continuos map and $c \in Y$ Is $\{x \in X | f(x)=c\}$ necessarily a closed subset of $X$? Attempt: I was thinking about a contradiction. Can this be a counterexample? ...
4
votes
1answer
236 views

On convergence of nets in a topological space

Let's consider a topological space that is not necessarily metrizable. Question: I wonder what is the motivation for defining convergence of nets in a topological space? What do we gain in working ...
7
votes
1answer
246 views

Convergence in the Box Topology.

Given the sequence in $\mathbb{R}^\omega$: $$y_1=(1,0,0,0\ldots), y_2=(\frac{1}{2},\frac{1}{2},0,0\ldots), y_3=(\frac{1}{3},\frac{1}{3},\frac{1}{3},0,\ldots),\ldots$$ I know that it converges in the ...
2
votes
1answer
381 views

limit point of the set $(-\infty, 0]\cup\{1/n:n\in\mathbb{N}\}$ with the subspace topology

Consider the set $(-\infty, 0]\cup\{1/n:n\in\mathbb{N}\}$ with the subspace topology. Then $0$ is an isolated point $(–2, 0]$ is an open set $0$ is a limit point of the subset ...
0
votes
1answer
128 views

Equivalence of metrics

I've got problem with following: $\phi : [0,\infty) \to \mathbb{R}$ is non-decreasing, concave function. Such that $\phi (0) =0$, and $\phi (u) >0$ for $u>0$. Prove that if $\phi$ is continuous ...
0
votes
1answer
19 views

Does consistency of a test imply continuity of a preorder?

Let $F$ and $G$ be two real-valued probability distributions, considered as cadlag functions with the uniform norm [not the Skorohod metric], and let $\mathbb{F}_n$ and $\mathbb{G}_m$ be empirical ...
1
vote
0answers
79 views

A theorem of a continuous map $f: S^1 \to S^1$

This is a theorem from my lecture notes: If the continuous map $f: S^1 \to S^1$ extends to a continuous map $F: B(0,1) \to S^1$ the $f$ is homotopic to a constant map. The proof just defines a ...
6
votes
2answers
183 views

Stone-Čech via $C_b(X)\cong C(\beta X)$

I am having some trouble constructing the Stone-Čech compactification of a locally compact Hausdorff space $X$ using theory of $C^*$-algebras. I did some search but could not find a good answer on ...
3
votes
2answers
5k views

Solution book of John Kelley's , J.Munkres's [closed]

I have so many difficult in solving problem in General Topology of John Kelley and Topology (second edition) of James R. Munkres. Does anyone know solution book of those? Just want to ask so many ...
2
votes
2answers
125 views

Which of the following sets are dense in $\mathbb{R}^2$ with respect to the usual topology?

Which of the following sets are dense in $\Bbb R^2$ with respect to the usual topology. $\{(x,y) \in\mathbb{R}^2:x\in \mathbb{N}\}$ $\{(x,y) \in\mathbb{R}^2:x+y \text{ is a rational number}\}$ ...
2
votes
1answer
225 views

topology-cofinite topolgy and co-countable topolgy

Let $T_1=\{U:X-U \text{ is finite for all of } X\}$. Then $T_1$ is the cofinite topolgy on $X$,where $X$ is an arbitrary infinite set. Then $T_1$ is not a Hausdorff space.Is it a regular space or a ...
3
votes
2answers
173 views

Base for a topology

I can prove that $(a,b)$ where $a,b$ are rational numbers form a countable base for the topology on $\mathbb{R}$. But, how to show that the collection $[a,b]$ where $a$ and $b$ are rational numbers, ...
6
votes
1answer
159 views

How does one see the topology of a Riemann surface from the graph (assuming one can picture $\mathbb R^4$)?

Given a function $f:\mathbb C\to\mathbb C$ which we will assume is analytic, we have an embedding $f\subseteq\mathbb C\times\mathbb C\cong\mathbb R^4$ of a surface. My question is with regards to how ...
1
vote
2answers
55 views

$f^{-1}$ and continuity

I really need help concerning this task: let $f : X \to \mathbb{R}^n$, $X\subseteq \mathbb{R}^n$, be a function with $X$ open. Show: $f$ is continuous $\iff f^{-1} (W) $ is open for every open $W ...
0
votes
1answer
40 views

neighborhood - simple but i need help

I have this set of complex numbers: $\{ 1-i , 2-i , 3-i \}$, and another set $$B:= \{ w \in \Bbb{C} \mid 0 \leq \mathrm{Re}(w)\leq 4 \land -2 < \mathrm{Im}(w)\leq0 \} \setminus \{ a+bi \in \Bbb{C} ...
4
votes
1answer
107 views

Is compact metric space separable in ZF?

Reference; http://www.samos.aegean.gr/math/kker/papers/CompactMetric.pdf The paper says "Compact metric space is separable" is unprovable in ZF$^0$( That is, ZF without axiom of regularity). And i ...
2
votes
1answer
961 views

Finite covering is compact Hausdorff iff base space is

I am in need of solution or tip for this question. I thank you. Let $ p: \widetilde X \to X $ be a covering space with $ p^{-1}(x) $ finite and nonempty for all $ x \in X$. Show that $ \widetilde X$ ...
3
votes
1answer
111 views

Topology and limit points

In my functional analysis homework, I had to prove something like this : Let $D_n \subset D_{n-1} \subset \dots \subset E$ where $(E, \mathfrak T)$ is a Hausdorff topological space and the sequence ...
0
votes
3answers
1k views

In a metric space, if a set is compact, then it is closed: improving proof

Let $(M,d)$ be a metric space. If $K\subset M$ is compact, then it is closed (and bounded). Proof Let's see that $M\setminus K$ is open. Let $x\in K$ $$\exists \varepsilon_1 (x), \varepsilon_2(x) ...
1
vote
2answers
162 views

Bolzano-Weierstrass proof correction

Bolzano-Weierstrass theorem Every sequence $\{ x_n \}_{n=1}^\infty$ bounded in $\mathbb R$ has, at least, a convergent subsequence. Proof Since $\{ x_n \}_{n=1}^\infty$ is bounded, then $\exists ...
7
votes
2answers
238 views

Relationships between AC, Ultrafilter Lemma/BPIT, Non-measurable sets

How is it possible to reconcile the following... In 1970, Solovay constructed Solovay's model, which shows that it is consistent with standard set theory, excluding uncountable choice, that all ...
4
votes
4answers
133 views

Show that the set given is closed

A question that I encountered which looks different than a normal open/closed sets proofs: Let $(E, d)$ be a metric space, let $f : E\to R$ be continuous and $a$ element of $R$. Show that the set ...
2
votes
0answers
39 views

Understanding intuitively that any loop $p:[0,1] \to S^1$ is end point preserving homotopic to a loop which doesn't change direction.

I'm trying to understand intuitively the proposition that: Any loop $p:[0,1] \to S^1$ is end point preserving homotopic to a loop which doesn't change direction. Surely a loop round a circle ...
0
votes
0answers
57 views

Lipschitz of multifunction of F

Let $F$ be a Lipschitz continuous multifunction from $\mathbb{R}^n$ to $\mathbb{R}^n$ with the Lipschitz constant $K$ and $$H=\sup\{\langle v,p \rangle|\thinspace v \in F(x)\}.$$ Prove that $K|p|$ is ...
1
vote
1answer
118 views

Question asked on the structure of open sets in $\mathbb{R}^n$

There was a question asked: An open subset $U\subseteq R^n$ is the countable union of increasing compact sets. There Davide gave an answer. Can anyone tell me how the equality holds, and the ...
0
votes
1answer
281 views

A continuous surjection that is not a quotient map

I was going through an old topology prelim, and encountered a question which I'm really not sure how I should work out. Here it is: Suppose we let $X = \mathbb{R} \times \{3,4,…\} \subset ...
4
votes
1answer
93 views

Let $X$ be a connected space, $f:X\to X$ continuous involution, $g:X\to\mathbb{R}$ continuous. Prove that $\exists$ $x\in X$ such that $g(x)=g(f(x))$.

This is a homework problem in an undergraduate topology class that I am ludicrously (and probably stupidly) stuck on. Any guidance would be appreciated: Let X be a connected space, $f:X\to X$ a ...
3
votes
2answers
238 views

In a metric space, compactness implies completness

Proposition Let $(M,d)$ be a metric space. If $K\subset M$ is compact, then $K$ is complete. Proof Let $\{x_n\}_{n=1}^\infty \subset K$ be a Cauchy sequence, then $$ \forall \varepsilon > 0 ...
5
votes
1answer
108 views

Different notions of local compatness and why are they implied by compactness?

There are several definitions of local compactness - from "every point has a compact neighbourhood", "every point has a base of compact neighbourhoods" to Hatcher's: "every neighbourhood contains a ...
6
votes
3answers
770 views

Set of limit points of a subset of a Hausdorff space is closed.

Let $X$ be a Hausdorff space and $A\subset X$. Define $A'=\{x\in X\mid x\text{ is a limit point of }A\}$. Prove that $A'$ is closed in $X$. Relevant information: (1.) Every neighborhood of a point ...
-2
votes
2answers
122 views

Is continuous function $f:X=A \cup B \rightarrow \{-1,1\}$ constant? [closed]

Let $A$ and $B$ be disjoint. Let $X$ be a topological space. Is every continuous function $f:X=A \cup B \rightarrow \{-1,1\}$ constant?
2
votes
1answer
591 views

Visually, why is the 2-sphere $S^2$ not contractible?

In topology, can someone please describe why the sphere $S^2$ is not contactable? Surely it can just 'shrink' to a point?
3
votes
2answers
128 views

sum of two connected subset of $\mathbb{R}$

$A$ and $B$ are two connected subset of $\mathbb{R}$ define $A+B=\{x+y:x\in A,y\in B\}$, then is $A+B$ also connected? naturally I was thinking about two disjoint connected subsets of $\mathbb{R}$, ...
3
votes
1answer
231 views

Proof that every open set in $\mathbb{R}^n$ is the union of …??

Can't figure this 2 part question out - I think the first part involves using open balls but I'm not sure. It's straightforward to prove this for $\mathbb{R}^1$ with disjoint segments but am a little ...
2
votes
1answer
35 views

Are 2 diagram homeomorphic?

Indeed there are many way to prove whether something are homeomorphic with each other. For the diagram below, it seems that they are not homeomorphic but i am not sure how to argue that.
2
votes
2answers
89 views

Is this mapping continuous?

If there is a mapping that is closed and open, is that enough to claim that that mapping is continuous? I can't really prove that or disprove that.
2
votes
1answer
197 views

Locally path connected space

Let $A\subset X$ and $p: E \to X$ be a covering space. Assume that $A,X,E$ are all locally path connected, path connected, Hausdorff. Suppose that $p^{-1}(A)$ is path connected. I want to show that ...
2
votes
2answers
266 views

Topology and locally closed subsets

Let $X$ be a topological space and let $A\subseteq X$. Supose that for each $x\in A$ there exists a neighbourhood of $x$, $V_x$, in $X$ such that $A\cap V_x$ is closed in $V_x$. Prove that ...
2
votes
2answers
186 views

Does every line segment in $\mathbb{R}^n$ contains a point having only rational coordinates?

Let $S$ be the subset of $\mathbb{R}^n$ consisting of all points which have only rational coordinates. I know that $S$ is a dense subset of $\mathbb{R}^n$. Is it true that every line segment in ...
7
votes
1answer
642 views

Product of connected spaces - Proof

I'm working on the "iff"-relation given by: $X=\prod_{i\in I}X_i$ is connected iff each $X_i$ non-empty is connected for all $i\in I$. I could prove the "$\Rightarrow$"-direction very easyly. I also ...
1
vote
1answer
181 views

Uniform convergence and complete metric space

Let $X$ be a metric space and $\{f_n\}$ be a sequence of functions such that $f_n:E\rightarrow X$. Suppose $f_n\rightarrow f$ uniformly on a set $E$ and $x$ is a limit point of $E$ and $\lim_{t\to x} ...
2
votes
3answers
2k views

Cover of (0,1) with no finite subcover & Open sets of compact function spaces

I just got back from my exam and these questions' solutions eluded me, it would be great to use the rest of my evening figuring these out... Q1: Find an open covering of the set $(0,1) \subset ...
1
vote
2answers
92 views

differentiability and compactness

I have no idea how to show whether this statement is false or true: If every differentiable function on a subset $X\subseteq\mathbb{R}^n$ is bounded then $X$ is compact. Thank you
2
votes
1answer
103 views

What is the topology on $[0,1]^2$ in the defintion of a homotopy?

I have a question about the definition of a homotopy between loops: Let $\alpha$ and $\beta$ be loops with base point $x$ in a topological space $X$. A homotopy from $\alpha$ to $\beta$ is a ...
5
votes
1answer
136 views

Proving that $O(n,m)$ is simply connected.

My question is the following: Under which conditions on given integers $n\le m$ is $$O(n,m) = \{A \in \mathbb R^{m\times n} : A^TA = \mathbf 1\}$$ simply connected? Does anyone know a reference for ...
2
votes
2answers
222 views

fixed point of homeomorphism and compactness of a complete metric space.

I need to know that the following statements if true or false: Every homeomorphism of $S^2\rightarrow S^2$ has a fixed point. Let $X$ be a complete metric space such that distance between any two ...
0
votes
1answer
154 views

a point of a subset in any topology either isolated or cluster point?

for metric space, we can say a point must be isolated point or cluster point. In generally, why it is not true? also, it is known that, a set is closed iff it is contained all cluster point. can we ...
4
votes
2answers
282 views

About covering maps and sections!

If $q: E\rightarrow X$ is a covering map that has a section $(i.e. f: X\rightarrow E, q\circ f=Id_X)$ does that imply that $E$ is a 1-fold cover?