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
42 views

Proof of a distance

I have one distance shown as an example in a book but I'm striving to demonstrate that it is effectively a distance. here it is said : let $U=\{z\in\mathbb{C, |z|=1}\}$ we can get a distance on $U$ ...
3
votes
1answer
56 views

Is Hausdorffness necessary for the classical ascoli theorem? [duplicate]

Munkres - topology p.278 I exactly followed the argument in the text, and I cannot find where I used hausdorffness. Where in the argument used Hausdorffness? The reason why I am asking is that the ...
4
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1answer
44 views

Closure and compactness of the set of real eigenvalues ​​of a real matrix.

Let $A$ be a part of $\mathcal{M}_n(\Bbb{R})$ and $B$ the set of real eigenvalues ​​of the matrix $A$. 1) Show that if $A$ is compact then $B$ is compact as well. 2) If $A$ is closed ...
9
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5answers
354 views

Why are box topology and product topology different on infinite products of topological spaces?

Why are box topology and product topology different on infinite products of topological spaces ? I'm reading Munkres's topology. He mentioned that fact but I can't see why it's true that they are ...
1
vote
1answer
425 views

Picturing Urysohn's Metrization Theorem and Urysohn's lemma?

In my Topological course we have this lemma. [Urysohn's lemma] Suppose that $X$ is a topological space. Then $X$ is normal if and only if, for each pair of disjoint closed subsets $A$ and $B$, there ...
0
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1answer
30 views

Open and closed equivalence relations

I am looking for canonical examples of open and closed equivalence relations, especially ones that are generated by a continuous functions. Intuitively I think that an open /closed continuous function ...
0
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0answers
11 views

Definition of Non-characteristic Manifold

What is the definition of non-characteristic manifold in the following context. "Since $ f (x, y)=0 $ is assumed to be a non-characteristic singularity manifold, we have $ f_{x}\neq 0 $." Thanks, ...
1
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3answers
49 views

Show that $Int(A)=X\setminus\overline{X\setminus A}$.

Let $A$ be subset in topological space of $(X,\tau)$, show that $\operatorname{Int}(A)=X\setminus\overline{X\setminus A}$. The definition of interior provided is "largest open set in A", which I ...
13
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9answers
1k views

Reference for general-topology

Though there are several posts discussing the reference books for topology, for example best book for topology. But as far as I looked up to, all of them are for the purpose of learning topology or ...
-2
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1answer
129 views

Every finite Hausdorff topological space is metrizable

Is it true that a finite, Hausdorff topological space is metrizable? That is, it is a topological space where the topology can be obtained from a metric.
15
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1answer
257 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
55 views

Is it true that $A$ is scattered?

Let $X$ be a (Hausdorff) topological space and for each ordinal $\alpha$ denote by $X^{(\alpha)}$ the $\alpha$th derivative of $X$ by the Cantor-Bendixson derivation (i.e., define transfinitely: ...
1
vote
1answer
23 views

On finite measurable space $X$, the whole of $L^p(X)$ is closed in $L^1(X)$ iff there is a const $C$ st $||f||_p\leq C||f||_1, \forall f \in L^p(X)$

On finite measurable space $(X, \mathcal{M}, \mu)$, the whole of $L^p(X, \mu)(p>1)$ is closed in $L^1(X,\mu)$ iff there is a const $C$ st $||f||_p\leq C||f||_1, \forall f\in L^p(X)$, iff both ...
2
votes
2answers
93 views

Show that the function $f:X\to \mathbb R$ defined by $f(x)=d(x,A)$ is continuous.

Let $d$ be a metric on $X$ and let $A$ be any arbitrary subset of $X$. Show that the function $f:X\to \mathbb R$ defined by $f(x)=d(x,A)$ is continuous. Let $p\in X$. We want to show that for any ...
2
votes
2answers
47 views

Is the product of $T_i$ spaces always a $T_i$ space?

I am doing some topology and wondering about the following. If $X_j$ is a $T_i$ space for some $i \in \{1,2,3,3.5,4\}$. Does it then follow that $\Pi_j X_j$ is again a $T_i$ space? I think for $i=2$ ...
2
votes
3answers
42 views

A homomorphism induces a continuous map from ${\rm Spec}(A') \to {\rm Spec}(A)$.

Let $A, A'$ be commutative rings with $1 \neq 0$. Let $h : A \to A'$ be such that $h(1) = 1$. Then $f: {\rm Spec}(A') \to {\rm Spec}(A)$ defined by $f(\mathfrak{p}') = h^{-1}(\mathfrak{p}')$ is ...
0
votes
1answer
50 views

Topological Manifolds & Covers

This problem is from John Lee's "Introduction to Smooth Manifolds" 1-4. Let M be a topological manifold, and let U be an open cover of M . (a) Assuming that each set in U intersects only finitely ...
1
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2answers
68 views

How can I show that the closure of the set $\Phi = \{1, 2, 3, 4,\ldots\}$ is the set itself?

If $\Phi \subseteq C(I)$, where $C(I)$ is the set of continuous real-valued functions on the interval $I=[0,1]$, and we define $\Phi = \{\phi_1, \phi_2,\phi_3,\ldots\}$ where $1=\phi_1$, $2=\phi_2$, ...
1
vote
2answers
48 views

Show that for any subsets $A,B\subset X$: (i):$d(A\bigcup B)\leq d(A)+d(B)+d(A,B)$ and (ii) $d(\bar A)=d(A)$

Let $d$ be a metric on $X$. Show that for any subsets $A,B\subset X$: (i) $d(A\cup B)\leq d(A)+d(B)+d(A,B)$ (ii) $d(\bar A)=d(A)$ I found this hard to prove because the diameter ...
4
votes
1answer
49 views

Product of T1 spaces is T1

I am trying to prove that the product of T1 spaces is also T1. Here is a proof, is it correct? $\{ X_i \}_{i \in I}$ are T1 $\Rightarrow$ $\prod_{i \in I} X_i$ is T1 Proof: Let $\bar{x} = ( ...
5
votes
1answer
152 views

Homeomorphisms between infinite-dimensional Banach spaces and their spheres

As I know Cz. Bessaga has proved that an infinite-dimensional Banach space is homeomorphic to its unit sphere. Unfortunately I do not have his book but I want to know is this theorem true without ...
1
vote
1answer
44 views

Show that $\overline{A\cap B}\subseteq\bar{A}\cap\bar{B}$

Let $A,B$ be subset of a topological space, show that $\overline{A\cap B}\subseteq\bar{A}\cap\bar{B}$. (The bar denotes closure) I have totally no clue, please give me some idea.
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0answers
51 views

Interior of a Dirichlet domain in a Riemannian manifold

Let $X\neq\varnothing$ be a complete connected Riemannian manifold. Suppose $G$ is a group of isometries of $X$, acting properly discontinuously on $X$. We assume there is a point $x_0\in X$ such that ...
3
votes
0answers
84 views

Intersections of two exponential curves in a plane

I am struggling to show that two exponential curves in $\mathbb{R}^n$ do not overlap except finite distinct points. Could anyone help me? Let $A\in\mathbb{R}^{n\times n}$ and ...
3
votes
3answers
96 views

Non-separable compact space

Off the top of my head, I can't think of a non-separable compact space. Can you provide a good example?
0
votes
1answer
36 views

Fixed point of continuous mapping between punctured disk

Let $X=B^2-\{a\}-\{b\}$, where $B^2$ is the unit disk on $\mathbb{R}^2$, $a, b$ are interior points of $B^2$. Is there a continuous map $f:X\rightarrow X$ which has no fixed point? Thank you.
4
votes
1answer
72 views

Whatever Happened to Nearness Spaces?

I came across this paper about Nearness Spaces. It seemed to be at the time (1970-80s) a promising approach to general topology via category theory. I have found no posts at all on stackexchange ...
5
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0answers
363 views

The Cantor Space as $\{0,1\}^{\mathbb{N}}$ and as $[0,1]$.

The Cantor-Space is defined as the space of all infinite binary sequences, i.e. the space $\{0,1\}^{\mathbb{N}}$. It has a natural metric, $$ d(x,y) = \inf\{ 2^{-|w|} : w \in pref(x) \cap pref(y) \} ...
2
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2answers
51 views

Find a CW complex with prescribed homology groups

A past qual question asks to construct a CW-complex $X$ with $H_0(X) = \mathbb{Z}$, $H_5(X) = \mathbb{Z} \oplus \mathbb{Z}_6$, and $H_n(X) = 0$ for $n\not= 0, 5$. One can build a CW-complex $Y$ by ...
7
votes
5answers
390 views

homeomorphism non-example

A homeomorphism is a continuous function between topological spaces that has a continuous inverse function. Can someone provide examples of a continuous function between topological spaces that does ...
3
votes
1answer
207 views

If radial projection is bijective then is it a homeomorphism?

Suppose $S$ is a regular surface in $\mathbb{R}^3 $ and $0\not\in S$. Now consider the radial projection $f: S\to\mathbb{S}^2$ given by $$f(x)=\frac{x}{||x||} \hspace{5mm}\mbox{ for all $x\in S$}$$ ...
11
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1answer
118 views

If $f\tau$ is continuous for every path $\tau$ in $X$, is $f:X\rightarrow Y$ continuous?

Let $X$ be a path connected space and $Y$ be a topological space. Let $f:X\rightarrow Y$ be a function such that for every path $\tau:\mathbb{I}\rightarrow X$ , $f\tau:\mathbb{I}\rightarrow Y$ is ...
1
vote
0answers
77 views

Ascoli-Arzela Theorem [duplicate]

Wikipedia gives the Ascoli-Arzela theorem for a Compact Hausdorff Space. Following through the proof sketch at the end of the linked section, I could not find where the Hausdorff property is used. The ...
2
votes
1answer
31 views

Homology groups of a simplicial complex

I have a question from a qualifying exam: let $X$ be the simplicial complex that consists of the 3-simplices $(v_1,v_2,v_3,v_4)$, $(v_3,v_4,v_5,v_6)$, $(v_1,v_2,v_5,v_6)$, where the $v_i$'s are all ...
1
vote
1answer
41 views

The existence of $f \in C^\infty(R^n)$ with $ f=0$ on closed $E$, otherwise $f>0$

This is problem 6.3 in 'Rudin's Functional analysis If $E$ is an arbitrary closed subeset of $R^n$, show that there is an $f \in C^\infty(R^n)$ such that $f(x)=0$ for every $x \in E$ and ...
5
votes
1answer
57 views

A maximal subset of $S^2$ with respect to a connectedness property

Let the set $A$ be a circle with a chord on the sphere $S^2$. Obviously $A$ has the following property: P: $\quad$ Any two points $a$ and $b$ of $A$ can be connected by a path that ...
0
votes
1answer
56 views

Density and convergence

I have a small question: Is it true that if the basis of a space $A$ is dense in a space $B$ ($B\subset A$) then if $u_n\rightarrow u$ in $A$ we have that $u_n\rightarrow u$ in $B$ ?
0
votes
0answers
61 views

Is $\sin (\mathbb N)$ dense in $[-1,1]$? [duplicate]

Let $\mathbb N$ be the set of positive integers, then is it true that $\sin (\mathbb N)$ is dense in $[-1,1]$ i.e. is it true that for every $x,y \in [-1,1]$ with $x<y$ , $\exists m \in \mathbb N$ ...
10
votes
4answers
566 views

Motivation for Topology study in Real Analysis

I'm an engineering student trying to work out some Real Analysis to learn how to write proofs (Needed for my PhD thesis) and just to rekindle my Calculus fires. From what I see, Real Analysis is the ...
5
votes
2answers
121 views

Is there a self-homeomorphism of the 2-sphere with exactly 3 fixed points?

I don't believe so, but I'm not sure how to prove it. The Lefschetz-Hopf theorem says in this case that the sum of the fixed point indices is 0 or 2 (since our map is a self-homeomorphism). My ...
0
votes
1answer
57 views

A detail in the proof of Jordan's theorem

The (usual) proof with homology first inductively shows that complements of embedded disks are acyclic. In doing so, Mayer-Vietoris is applied, and this assumes that their complements in the sphere ...
2
votes
1answer
60 views

Definition of disc and open ball

I have the following definitions in my notes for arbitrary discs and open balls - $$D^n = \{x \in \mathbb{R^{n+1}}: ||x|| \le 1\}$$ $$B^n = \{x \in \mathbb{R^{n+1}}: ||x|| < 1\}$$ The ...
10
votes
1answer
526 views

Complement of a totally disconnected compact subset of the plane

Let $E \subset \mathbb{C}$ be compact and totally disconnected. Is there an elementary way to prove that $\mathbb{C} \setminus E$ is connected?
0
votes
0answers
32 views

Disconnecting using totally disconnected sets [duplicate]

Let $X$ be $[0,1]^2$ and $S\subset X$ a totally disconnected subset. Is it true that $S^c$ is always connected? If it is false, what can we say when $X=[0,1]^n$?
2
votes
1answer
56 views

A question of topology.

If S is a subset of $\hspace{0.1cm}$$[0,1]\times[0,1]$$\hspace{0.1cm}$ such taht one point of the ordered pair is rational and the other is irrational or both are irrationals,then which of the ...
13
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3answers
4k views

Topology exercises

Can anyone suggest a collection of (solved) exercises in topology? Undergrad level, as a companion to Dugundji's Topology (although excellent it doesn't provide the solutions to the problems). ...
0
votes
1answer
26 views

How do I calculate the Jacobian matrix of the transformation of a 1-m manifold to a chart (topology question)?

What I want to do is take a 1-m manifold (something like a circle), and transform a subset of that manifold into a chart. I want to represent that function from manifold to chart with a 1 x 1 matrix, ...
2
votes
1answer
86 views

Is there a nice/clever way to visualize $\mathcal{S}\times \mathbb{R}^2$?

The (velocity) phase space of a double pendulum can be seen as the tangent bundle of its configuration space ($\mathcal{S}^1\times\mathcal{S}^1$), that is: ...
2
votes
1answer
83 views

A question on the closure of a set

I do not know how to prove the following. However, it seems to be true when I set A=rational numbers and B=irrationals. Any hint will be helpful. ...
0
votes
0answers
42 views

Finding a homeomorphism between these two balls

Let $u_1,u_2,u_3 \in \Bbb C$ be the cubic roots of unity. Define two norms on $\mathbb{C}^2$, $$\Vert (x,y) \Vert_1 = \sqrt{\vert x \vert^2 +\vert y \vert^2} \ \text{and} \ \Vert (x,y) \Vert_2 = ...