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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Compactness and convergence

Let $U$ be a subset of $\mathbb{R}^n$, and suppose that $U$ is not bounded. Construct a sequence of points $\{a_1, a_2, \ldots \}$ such that no subsequence converges to a point in $U$, then prove this ...
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4answers
83 views

How should I think of an open vs. closed set?

I've been studying introductory topology for a little bit now. I came across this video which explains open sets in a way I have never thought of. Even though the video is pretty elementary, I didn't ...
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3answers
2k views

Example to show the distance between two closed sets can be 0 even if the two sets are disjoint

Let $A$ and $B$ be two sets of real numbers. Define the distance from $A$ to $B$ by $$\rho (A,B) = \inf \{ |a-b| : a \in A, b \in B\} \;.$$ Give an example to show that the distance between two closed ...
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0answers
27 views

How to show a map is a homeomorphism?

I have calculated two of the properties of homeomorphism. Where I have found the bijective mapping and showed that $f$ is continuous. However i am not sure how to show that $f^{-1}$ is continuous?
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1answer
24 views

The special unitary group SU(2) is homeomorphic to the 3-sphere

The special unitary group is defined as $$SU(2) = \{A\in M_{2,2}((C) \mid A\bar A^T=I\}$$ Show that this is homeomorphic to the $3$-sphere $$\mathbb{S}^3 = \{(a,b\in \mathbb{C}^2\mid ...
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1answer
18 views

Join of closed embeddings is a closed embedding

An exercise from James's book General Topology and Homotopy Theory asks the reader to prove that if $\phi_1:X_1 \to Y_1$ and $\phi_2:X_2 \to Y_2$ are closed topological embeddings, then $\phi_1 * ...
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0answers
66 views

$A \subset \Bbb R$ such that $A$, $clA$, $int(A)$, $cl(int(A))$, $int(clA)$ are pairwise distinct

Do there exist subsets with internal closures $A$ of $\mathbb R$ such that $A$ , $\bar A$ , $A^0$ , $(\bar A)^0$ , $\overline{A^0}$ are pairwise distinct ? I found an example from a book that such a ...
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0answers
15 views

Metric Spaces open function

Let X and Y are two metric spaces and f is bijection function from X to Y . Prove that f is open iff f is closed
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1answer
39 views

Show that $p:SO_3 \to\mathbb S^2 $ defined as $p(A)=Ae_1$ is a fibre bundle

Show that $p:SO_3 \to\mathbb S^2 $ defined as $p(A)=Ae_1$ is a fibre bundle. I know that $SO_3$ acts on $\mathbb S^2$ transitively saying that $p$ is onto.I have a problem with local ...
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3answers
28 views

Topology space closed set [on hold]

Why $A =\{ (x,1/x), 0\neq x\in R \}$ is closed in $R^2$?
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0answers
50 views

How to prove whether a map is one-to-one using disconnected property

I found the same Question here Prove that $S: I \rightarrow \Sigma_2'$ is one-to-one. But it is unsolved. Moreover, I do not understand the answer in the Question. Definition 1 : $\Sigma_2 = ...
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0answers
15 views

Topological structure/graph from a paper

This question is based off a paper titled "On designing heteroclinic networks from graphs." I'm having a difficult time visualizing something "drawn in 4-dimensions" projected down to a 2-dimensional ...
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1answer
21 views

Difficulty with a differentiation of measures proof

This shows up in a proof about differentiating measures. I'm having trouble figuring it out: For any $x \in \mathbb{R}^n$, let $\mathcal{C}_r(x)$ denote the set of open cubes with diameter less than ...
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3answers
63 views

Equivalence of countable choice for subsets of the reals and “second countable $\implies$ Lindelöf”

Looking for a proof that in a second countable space every open cover has a countable subcover -- i.e. every s.c. space is a Lindelöf space -- I bumped into this question. That answered my question. I ...
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2answers
38 views

show the function is a homeomorphism?

Show that $f:(-1,1)\to\mathbb R, f(x)=\frac{x}{1-x^2}$ is a homeomorphism. In order for this function to be a homeomorphism it needs to be bijective and its inverse needs to exist and be ...
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1answer
16 views

How can this function be considered to have a saddle node bifurcation?

Say I have the function $f(x,\mu) = (1 + \mu)x − x^2 − 0.1$. By definition a Saddle Node bifurcation occurs if: $f_{\mu_0}(0) = 0$ $f'_{\mu_0}(0) = 1$ $f''_{\mu_0}(0) \neq 0$ $\frac{\delta ...
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1answer
41 views

Let $g: S^2 \to S^2$ be continuous and $g(x) \neq g(-x)\ \forall x$. Prove that $g$ is surjective.

Let $g: S^2 \to S^2$ be continuous and $g(x) \neq g(-x)\ \forall x$. Prove that $g$ is surjective. The hint that if $p \in S^2$, then $S^2 - \{p\}$ is homeomorphic to $\mathbb{R^2}$. It is ...
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1answer
33 views

If $\tau_1, \tau_2,\tau_3$ then which are correct?

Let, $\tau_1, \tau_2,\tau_3$ be three topologies on a set $X$ such that $\tau_1 \subset\tau_2\subset\tau_3$ and $(X,\tau_2)$ be compact $T_2$ space. Then , (A) $\tau_1=\tau_2$ , if $(X,\tau_1)$ is ...
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0answers
29 views

Topology, maps, continuity

I know how to write all the maps, however, how would i find out which of those are continuous.
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1answer
32 views

Question on Furstenberg topology on Z and P subspace of primes

Hi all I was given this question: I have Z (the integers) with the Furstenberg topology on it, i.e. the topology induced by non constant arithmetical progressions presented here, and I am asked to ...
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0answers
12 views

projective plane and topology [duplicate]

Gluing the Mobius strip with a disk altogether along their boundaries gives $\mathbb RP^2$. Please give me some hint or explanation how to solve it, why it is true. And give a topology on this ...
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0answers
18 views

What is the Topology of Cantor set as the subset of real line with standard euclidean topology? [duplicate]

It may be an odd question. But, I particularly want to know the open's of cantor set with inherited topology of real line.
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1answer
47 views

Prove that the given function is bounded, but not continuous, on the the given region

Let , $$f(z)=\begin{cases}e^{(−1/z)} & \text{ if } z\not=0\\0 & \text{ if }z=0\end{cases}$$ prove that the function is bounded, but not continuous, on the half circle $0\le|z| \le1, ...
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2answers
56 views

$f(z)=\frac{1}{z}$ has an antiderivative on any simply connected domain

Prove that the function $f(z)=\frac{1}{z}$ has an antiderivative on any simply connected domain of $\Bbb C$ which does not contain zero. Also prove that this function does not have an ...
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0answers
48 views

transformation of a folded piece of paper!!!

This is a question in the book Real Mathematical Analysis by Charles Chapman Pugh and I don't know how to face it! : Fold a piece of paper in half. (a) Is this a continuous transformation of one ...
2
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2answers
54 views

Proving that if a set is both open and closed then it is equal to the real numbers

Prove that if $A$ is both open and closed then $A = \mathbb{R}$ also as one suggested let $A \neq \emptyset$ You may use what ever definition of open and closed you would like, just avoid going into ...
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1answer
76 views

How many topologies exist on a finite set?

In my topology class we are asked to list all topologies on a $3$ element set. I have found $29$ and this should be the correct result. Now I wonder whether there is some formula that determines this ...
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0answers
12 views

Homomorphisms between countable spaces and Euclidean spaces?

Is there some place to start reading about homomorphisms between countable (discrete) spaces and Euclidean spaces or $l_2$? I know it is a rather general question, but I am not sure what I am looking ...
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0answers
21 views

How to prove that every countable compact Hausdorff space is homeomorphic to a well-ordered set with its order topology? [duplicate]

I encountered this problem in a textbook: Let $X$ be a countable compact Hausdorff topological space, I was asked to prove that $X$ is always homeomorphic to a (necessarily countable) topological ...
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1answer
16 views

Is there a lower bound for the maximal number of separated sets?

Let $(X,d)$ be a metric space and $T\colon X\to X$ uniformly continuous. A set $E\subset X$ is said to be $(n,\varepsilon)$-separated if for any distinct $x,y\in E$ there is a $0\leq j< n$ such ...
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0answers
19 views

extend $f: S\longrightarrow \mathbb{R}$ to $f^* : cl(S)\longrightarrow \mathbb{R} $ [duplicate]

if $f: A\longrightarrow B$ and $g:C\longrightarrow B$ such that $A\subset C$ and for each $a\in A , f(a)=g(a)$ then f extends to g. Assume that $f: S\longrightarrow \mathbb{R}$ is uniformly continous ...
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0answers
68 views

$X$ is A-space iff the frontier of any closed set in $X$ is compact.

Hi everyone I have troubles with the following proposition: Definition: We say a metric space $(X,d)$ is an A-space iff every Hausdorff image of $X$ under a closed continuous map is metrizable. ...
3
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1answer
8 views

Does “equalisers always closed” imply $T_2$?

Is there a non-$T_2$ space $(X,\tau)$ with the following property? For all topological spaces $A$ and continous maps $f,g:A\to X$ the set $\{a\in A: f(a) = g(a)\} \subseteq A$ is closed.
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1answer
32 views

Given two closed sets $A,B$, show that there exists some of them which contains all distances

Suppose that $A,B\subseteq[0,1)$ are closed, and that $A\cup B=[0,1)$. Show that there exists a set $C\in\{A,B\}$ such that given an $x\in[0,1)$, $C$ contains two points $p$ and $q$ such that ...
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1answer
22 views

Prove that exist bijection between inverse image of covering space

Let $B$ be path-connected and $p:E\to B$ covering map (with $E$ as covering space). Prove that $\forall a,b\in B$ exist 1-1 injection correspondence between $p^{-1}(a)$ and $p^{-1}(b)$ I thought ...
4
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1answer
65 views

If $f:S^1\to S^1$ doesn't have any fixed point then it is homotopic to the identity

How to show that every continuous function $f:S^1\to S^1$ without fixed points is homotopic to the identity? (without using homology nor the concept of degree).
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1answer
84 views

Klein bottle and real projective two-space

I've been looking at equivalence relations on the unit square: $[0,1] \times [0,1]$ that give rise to various surfaces such as the m$\ddot{\mathrm{o}}$bius strip, but I'm not too sure about the Klein ...
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2answers
49 views

How to prove the group of automorphisms of $S^1$ as a topological group is $\mathbb Z_2$?

The title basically says it all. How does one prove the group of automorphisms of $S^1$ (the unit circle in $\mathbb C$), as a topological group, is $\mathbb Z_2$? I was surprised not to find the ...
2
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1answer
27 views

Show that an open subset $U \subset \mathbb{H^n}$ is open in $\mathbb{R^n}$ iff $U \cap \partial\mathbb{H^n}=\phi$

Consider the $n-$dimensional closed half-space $\mathbb{H^n}=\{x \in \mathbb{R^n}|x_1 \le 0 \}$ and let $\partial \mathbb{H^n}=\{x \in \mathbb{H}|x_1=0\}$ be its boundary. Show that an open subset $U ...
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1answer
36 views

Number of path connected components

Is it possible to give an explicit characterization of compact subsets of $[0,\infty)$. Is it true that given any compact subset $K \subseteq [0,\infty)$ then $[0,\infty) \setminus K$ has only one ...
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1answer
34 views

Use Least Upper Bound to show that $\mathbb{R}$ is completee

Use Least Upper Bound to show that $\mathbb{R}$ is complete. The following is the proof I did, and it's slightly different from what I see from the book. Can someone check if I'm missing ...
2
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1answer
19 views

Example of metric continuous with respect to another metric but generating different topology

Take, say, the standard 2-sphere $S^2$. Equip it with some metric $d$; this metric will generate a topology that may or may not coincide with the standard Euclidean topology. In the case it does, ...
2
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0answers
55 views

Surgery to unlink $S^1$ and $S^2$ in $S^4$

Let us start with a $S^1$ and a $S^2$ are linked in $S^4$. Can I unlink the $S^1$ and $S^2$ by doing some surgery (with certain constraints described below, and let us say both $S^1$ and $S^2$ ...
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1answer
40 views

Why every algebra on finite set is a topology

How can I prove that every algebra on finite set is a topology on this set And if the set is infinite how can give me an example algebra but it isn't topology
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2answers
25 views

Compactness and Hausdorffness with different topology

Here is the question (Munkres pg. 170): Show that if $X$ is compact Hausdorff under both $\mathcal{T}$ and $\mathcal{T}'$, then either $\mathcal{T}$ and $\mathcal{T}'$ are equal or they are not ...
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1answer
26 views

Closure of bounded set is bounded? Topological space

Let $X$ be topological space, and $A \subseteq X$ that is bounded. Is the closure of $A$ also bounded? This is true if $X$ is topological vector space, but is it if $X$ is only topological?
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1answer
81 views
+50

Show that $\bar A = A \cup [(0,0), (0,1)]$

In $(\mathbb R^2, ||\cdot||_{\infty})$, let: $A_0 = ]0,1] \times \{0\}$ $A_n = [(\frac{1}{n}, 0), (\frac{1}{n}, 1)]$ for each $n \ge 1$. $A = \cup_{n=0}^{\infty} A_n$ It is required to prove that: ...
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3answers
851 views

Are all finite sets closed?

say $X=\{a,b\}$ be a set. The following is a topology on $X$. $\tau=\{\{ a\}, \{a,b\}, \varnothing\}$ Then $b$ is a limit point of $a$, as all open sets $(\{a,b\})$ intersect $\{a\}$ at points other ...
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1answer
29 views

Is this topological space metrizable?

Let $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ carry the product topology $\tau$. Surely, $(X,\tau)$ is compact (Tychonoff). But is $(X,\tau)$ metrizable and if yes - why? Is there a metric that ...
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0answers
18 views

Using Bounded Operator sequence Theorem

Let E$\subset L_1$ be a set of fourier series functions $e_n(t)=e^{int}$ for $n \in Z$. What is meant by saying to prove $Ge_n$ is a scalar multiple of $e_n$ and it is continuous? How can we prove it? ...