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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23 views

On the completeness of Weak Operator Topology

Let E,F be any two Banach space and let $\mathcal{B}(E,F)$ be the space of all bounded linear operators from E to F. I can show that this space is a complete space with respect to the norm and strong ...
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1answer
25 views

Which of the following are compact I need Hint…

Which of the following are compact? $\{(x,y) \in \mathbb{R}^2 :(x-1)^2+(y-2)^2=9\} \cup \{(x,y) \in \mathbb{R}^2: y=3\}$. 2.$\{(\frac{1}{m},\frac{1}{n}) \in \mathbb{R}^2:m,n\in \mathbb{Z}-\{0\}\} ...
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1answer
160 views
+50

$m+ni+k\lambda,\,\Re(\lambda),\Im(\lambda)\notin \mathbb{Q}$ is dense in $\mathbb{C}$!

As said in the comments below, it's needed to suppose $\{1,\Re(\lambda),\Im(\lambda)\}$ linearly independent over $\mathbb{Q}$, otherwise the result is false, according to Christian's example. ...
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0answers
32 views

Two self maps $f,g:S^n\to S^n$ are homotopic if there is no $x\in S^n$ with $f(x)=-g(x)$

What I want to prove: Let $f,g:S^n\to S^n$ be continuous. If there is no point $x\in S^n$ with $f(x)=-g(x)$ then $f\sim g$, then . I am using this as a lemma to prove a slightly bigger result: ...
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1answer
27 views

Prove that the function $I:(X,d^*)\to(R,d)$ is continuous

Let $X$ be the set of continuous functions $f:[a,b]\to R$. Let $d^*$ be the distance function on $X$ defined by $$d^*(f,g)=\int_{a}^{b}|f(t)-g(t)|dt$$ for $f,g\in X$. For each $f\in X$, set ...
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22 views

When the boundary of a manifold is orientable?

I am not sure whether the boundary of some manifold is definitely a manifold, but let's assume it is anyway. Then in what case the boundary is an orientable manifold. Maybe when the manifold can be ...
0
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1answer
26 views

Connecting boundary definitions

I've noticed that there are several definitions of what a boundary of set is. However, I have problems with connecting them together. If I say define the boundary as $\partial A = \bar{A} \backslash ...
4
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1answer
37 views

Which surface is homotopy equivalent to $\Bbb{R}^4$ minus the planes $x=y=0$, $z=w=0$?

In completing an exercise I have shown that $\Bbb{R}^3$ minus the axes $x=0$, $y=0$, and $z=0$ is homotopic equivalent to the cube graph $Q_3$. To visualize this, $\Bbb{R}^3-0$ is homotopy equivalent ...
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25 views

What is the $\epsilon$ neighborhood of a subset in $\mathbb R^2$ in $\mathbb R^n$

Let X denote the subset $(-1,1) \times 0$ of $\mathbb R^2$ and let U be the open ball B(0,1) in $\mathbb R^2$ which contains X. Show that there is no $\epsilon > 0$ s.t the $\epsilon$-neighborhood ...
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26 views

Showing $D = \{ f : |f(x) - f(y)| \leq \sqrt{|x-y|} \}$ where $D \subseteq C[0,1]$ is not bounded

Given $D = \{f: |f(x) - f(y)| \leq \sqrt{|x-y|}\}$ where $D \subseteq C[0,1]$, I am asked to show this set is not compact. To do so, I must show the set is not bounded (specifically not uniformly ...
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0answers
19 views

$d:\mathbb N \times \mathbb N \to \mathbb R \ d(m,n)=0:m=n ; d(m,n)=1-\frac{1}{m+n}:m \neq n. $ Question convergence of sequences $(x_n)(y_n)(z_n)$

$d:\mathbb N \times \mathbb N \to \mathbb R \ d(m,n)=0:m=n ; d(m,n)=1-\frac{1}{m+n}:m \neq n. $ Question convergence of sequences $(x_n)(y_n)(z_n)$ and prove that $B_n=\{m \in \mathbb N : d(m,n)\leq ...
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1answer
24 views

Adding an isolated point to a Borel space

I have a Borel space $S$, which is basically a Borel subset of a Polish space. I want to add an isolated point $\alpha$ to $S$. Let $\overline{S}=S\bigcup \{\alpha\}$. Can I say that $S$ is clopen in ...
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1answer
21 views

What is the closure of an open ball $B_X(\mathbf{a},r)$ in $X=\mathbb{R}^n$?

Suppose we have the open ball $B_{X}(\mathbf{a},r)$ and the closed ball $\bar{B}_{X}(\mathbf{a},r)$ of radius $r$ about $\mathbf{a}\in\mathbb{R}^n=X$ with the Euclidean metric $d_2$. What is the ...
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23 views

Regular closed sets in a subspace of a topological space

Suppose $\langle X,\mathscr{O}\rangle$ is a locally compact and Hausdorff topological space and $\operatorname{RC}(X)$ is its algebra of regular closed sets (i.e. these that are equal to the closure ...
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2answers
43 views

How can I show $R^n$ is dense in $S^n$?

How can I show $R^n$ is dense in $S^n$? I wanted to show $S^n$ is compactification of $R^n$. for this I need $R^n$ is not compact, for this there is no problem, and $S^n$ is compact, I did it with ...
0
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1answer
30 views

Continuous functions between topological spaces and their homotopy equivalence relations

Let $A,B,C$ be topological spaces and $\alpha,\alpha':A\rightarrow B$ continuous and $\beta,\beta':B\rightarrow C$ be continuous. Let $\sim$ be the homotopy relation (which I know/can use to be an ...
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2answers
58 views

A complicated lemma from Munkres's Topology

I don't understand the following lemma of the book Topology by J Munkres : 1- If a set $A$ with the mentioned properties exists there must be an example for it and so, it could help to understand ...
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1answer
22 views

Partition into open sets and their boundary

Let $\tau$ be a topology on $X$. Let $\omega \subseteq \tau$ be a set of pairwise disjoint, open subsets of $X$. Let $Y := \bigcup\limits_{Z \in \omega} Z$. Question: If $X = Y \cup \partial Y$, ...
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1answer
37 views

With justification, determine whether or not the following space is compact.

The space in question is the Hausdorff topological space with base β: β = {U(a, b) : a, b ∈ Z, b > 0}, where U(a, b) = {a + kb : k ∈ Z} . (I have confirmed that this in fact a base of a Hausdorff ...
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3answers
41 views

Is the closure of the set of all irrational rotation maps on $S^1$ dense in $Homeo(S^1)$?

I study about rotation maps on circle, and I have a question. Let $Homeo(S^1)$ be the set of all circle homeomorphisms with sup-metric $d(f,g)= \sup \{ d(f(x),g(x)| x \in S^1 \}$, and rotation map ...
1
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1answer
19 views

Show that the set of symmetric positive definite matrices with determinant +1 is connected

I tried constructing a path $C(t) = tA + (1-t) B$ for any two matrices $A,B$ in the set. Clearly, $C(t)$ is still positive definite and symmetric for $t \in [0,1]$. But I ran into issues in showing ...
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0answers
28 views

Measurability of integrals with respect to different measures [closed]

Let $Y$ be a locally compact Hausdorff topological space (further assumptions like metrizability, separability, etc., may be added if necessary) and let $\mathscr Y$ denote the Borel $\sigma$-algebra ...
0
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0answers
12 views

Computing boundary homomorphisms in cellular chain complex of lens spaces

I am working on my own through Hatcher's book and I am having trouble while understanding the computations of cellular homology for the CW-complex structure of Lens spaces. It is on page 145. I upload ...
0
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1answer
37 views

Consequence of Brouwer / No Retraction

I am trying to prove a consequence of the Brouwer Theorem and/or the No Retraction Theorem and I think it must be simple I am just not seeing the point. The result is: If $f:D^2\rightarrow D^2$ is ...
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0answers
26 views

understanding a proof for uniform convergence of Fourier features

See part of the proof. In the proof, $\tilde{f} := \tilde{z}(x)'\tilde{z}(y) - k(x,y)$, the vector product is the approximation of a shift-invariant kernek $k(x,y)$. What is not clear to me is the ...
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2answers
31 views

Construct a homemorphism $\phi : T^2/A \rightarrow X/B $

Construct a homemorphism $\phi : T^2/A \rightarrow X/B $ $T^2=S^1 \times S^1$ and $A \subset T^2$ is given by $A=S^1 \times\{1\}$. $X=S^1 \times [-1, 1]$ and $B = S^1 \times\{-1, 1\}$. ...
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5answers
76 views

Cover $(0, +\infty )$ by open sets

Cover $(0, +\infty)$ by open sets $U_\alpha$ such that for any $\epsilon > 0$ there are points $x, y \in (0, +\infty)$ with $|x-y|<\epsilon$, not both belonging to the same $U_\alpha$ The ...
0
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1answer
33 views

Homology group of Klein bottle via Mayer Vietoris. Explanation of “ Since the boundary circle of a Mobius band wraps twice around the core circle ”

My qeustion about first homology group of Klein bottle via Mayer Vietoris sequence . i have exact sequence below $ 0 \xrightarrow{} H_1( S^1) \xrightarrow{(i,j)} H_1(M) \oplus H_1(M^\prime ...
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0answers
50 views

Compute Euler characteristic $\chi(S^2 \times S^3)$

This is a question from an undegraduate Topology course Compute the Euler characteristic $\chi(S^2 \times S^3)$ For topological polyhedra $\chi(X \times Y)=\chi(X) \times \chi(Y) \implies ...
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3answers
41 views

Prove that half-open sets in $\mathbb{R}$ are measurable

Self-learning these concepts, so please be tolerant with imprecise terminology... Defining the standard topology on the real line $\mathbb {R}$ as all the open intervals, a Borel $\sigma$-algebra is ...
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0answers
25 views

Explicit formulas for meets and joins of uniform spaces

I want explicit formulas for meets and joins (and finite meets and joins) for sets of uniform spaces (where uniformities are ordered by inclusion). And also for proximity spaces. I am also ...
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0answers
14 views

Union of the unit circle $S^1$ and the curve is connected but not path-connected

Prove that the union of the unit circle $S^1$ and the curve $W=\{(x, y) \in \mathbb{R^2} | x=(1-e^{-t})cost, y=(1-e^{-t})sint, t \geq 0\}$ is connected but not path-connected A connected space is ...
1
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1answer
41 views

sphere-filling curve

Let $S^2$ denote the $2$-dim sphere in $\mathbb R^3$. I am interested in finding a space-filling curve, i.e. a map $\varphi: [0,1]\to S^2$ that is continuous and onto. We know that there is such a ...
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0answers
30 views

Which of these graphs are homotopic but not homeomorphic?

I am struggling with the 'homotopic' part of the question Which of these are homotopic but not homeomorphic? The number of vertices of degree $\neq 2$ is a topological invariant, thus is true ...
3
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1answer
43 views

For every group $G$ there is a $2$-dimensional cell complex $X_G$ with $\pi_1(X_G)\cong G$.

I am reading Allen Hatcher's Algebraic Topology, and am trying to understand the proof to corollary 1.28: For every group $G$ there is a $2$-dimensional cell complex $X_G$ with $\pi_1(X_G)\cong ...
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0answers
36 views

Show that minimum and maximum are contained in $V(\mathcal{R})$/ Stones' axiom

Let $\mathcal{R}\subset\mathcal{P}(\Omega)$ be a ring for some set $\Omega$. Consider $$ V:=V(\mathcal{R}):=\left\{\sum_{i=1}^n\alpha_i1_{A_i}: \alpha_i\in\mathbb{R}, A_i\in\mathcal{R}, ...
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1answer
22 views

CW construction of Lens spaces Hatcher

I am working through Hatcher's book and I am having trouble while understanding the CW-complex structure of Lens spaces. It is on page 145. He proves it constructing it in an inductive process. I ...
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0answers
37 views

the point set is nowhere dense in $X$

I am trying to show for a metric space $X$, a set $\{x\}$ consisting of a single point is nowhere dense. I have proven it by showing $[{\{x\}}]^o = \emptyset$ where $[A]$ is the closure of the set $A$ ...
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1answer
28 views

$T_0$ is equivalent to $T_1$ in a topological group

I need to prove that the separation axiom $T_0$ is equivalent to $T_1$ in a topological group $G$. $T_1$ implies $T_0$. So I only need to prove $T_0$ inplies $T_1$. What I have tried is: ...
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0answers
19 views

Fundamental Groups of a Simplicial Complex and the Underlying Space

Now I know how the fundamental group of a Simplicial Complex is defined, as well as that of a Toplogical Space. Could someone explain the process of how we prove that for a simplicial complex $X$ , ...
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0answers
19 views

2-Cocycles and Automorphisms

hopefully a short question, which can be answered by someone who is more into that topic than I am. Suppose we have a $\mathbb{Z}_2$ valued 2-cocycle $\epsilon$ defined on a root lattice $\Phi$ of a ...
0
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1answer
40 views

If $\Omega$ is an open set of $\mathbb{C}$, $f$ constant on each connected component then $f$ is continuous

Let $\Omega$ be an open set of $\mathbb{C}$ and $f$ a constant function on each connected component of $\Omega$. I need to proof that $f$ is continuous. I've tried using that connected components ...
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34 views

Why this doesn't determine a torus?

Why the diagram on Figure 3.6 doesn't determine a torus? Do we need at least three rows of rectangles?
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32 views

Canonical metric on the suspension of a metric space

Let $(X,d)$ be a metric space. Is there a metric $d'$ on the (unreduced) suspension $\Sigma X = (X\times[-1,1])/\sim$ of $X$ such that $d'$ restricts to $d$ on $X\times \{1/2\}$? Further, we would ...
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1answer
35 views

Isolated and discrete points

In the space of integers with usual topology, is it true that 1) every point is isolated point for any set 2) no point is limit point for any set. In real space and space of rationals 1)no point is ...
0
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1answer
21 views

Example of dense set in space of integers

Let X be the space of integers where topology T is class of sets $\emptyset, \{1\}, \{1,2\}, \{1,2,3\},\{1,2,3,4\} \cdots, X$. Is it correct that each open set except $\emptyset$ is dense in space of ...
0
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1answer
57 views

Closure of a set in topological space

Closure of a set $A$ in topological space $X$ is defined as intersection of all closed supersets of $A$. Can we say that $Closure(A)= \bar{A}$ such that $(\bar{A})'=\bigcup B$ where $B$ is the class ...
0
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1answer
32 views

Let $(x_\alpha)_{\alpha\in J}$ be a net in a topological space $X$. Show that if $J$ is a finite set then the net converges? …

Let $(x_\alpha)_{\alpha\in J}$ be a net in a topological space $X$. Show that if $J$ is a finite set then the net converges? ... Why does it have to converge? I don't understand why with nets this is ...
2
votes
1answer
64 views

Why is a line not quasi-isometric to a plane?

I really don't know how this is true, because I need to prove that there are no mappings from a line to a plane that is quasi-isometric. I think my starting point is probably wrong. Could you give a ...
3
votes
1answer
35 views

H-space multiplication question (homotopy)

Let $(X,x_0)$ be a $H$-space with multiplication $\mu:X\times X\to X$. Let $e$ denote the constant map $I^n\to x_0$. Is it true (and why) that $\begin{cases} ...