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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7 views

Compactification of a straight line

Like in the case of mapping a infinite-plane to a sphere (Riemann Sphere), I can understand, that I can map the infinite line ($-\infty,\infty$) to a circle. Secondly, I can also map a finite line ...
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26 views

Hausdorff, regular and separable space

Let be $X$ a topological space such that X is a Hausdorff, regular and separable space. If $U\subseteq X$ is open such that $U=int(cl(U))$, and $E\subseteq X$ is a countable dense set, I need to prove ...
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0answers
11 views

Linking of $S^p$ and $S^q$ in the $\mathbb{R}^d$ space

Can we have a nontrivial linking of a $S^p$ sphere and a $S^q$ sphere in the $\mathbb{R}^d$ space (or in the ${S}^d$ space)? I suppose that it can happen only if $p+q<d$. For example, we can have: ...
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2answers
25 views

Show that the Möbius band has its central circle $C$ as a deformation retract

I have started this problem by using the planar representation of the Möbius band and noted that a line down the middle is probably what is meant by the central circle, since travelling from top to ...
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0answers
7 views

Homology group and homotopy group of the standard twin

Given a 4-sphere, if we cut out a solid 3-torus $B^2 \times S^1 \times S^1$ from a 4-sphere $S^4$ (with an unknotted torus), the remained exterior is called "the standard twin," say $M$. What are ...
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1answer
12 views

Show that the lemniscate is compact as a subspace topology . [on hold]

define a map $f| (- \pi\ ,\pi) \to \mathbb{R^2} , f(x)=( \sin(2x),sin(x))$ The image of $f$ is the lemniscate (a figure 8 curve). Show that this image is compact in the subspace topology. This means ...
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18 views

Wot topology on $B(H)$ is not metrizable

Let $H$ be a infinite dimensional Hilbert space and $B(H)$ be the space of bounded and linear operators on $H$. I know that weak operator topology (wot) and strong operator topology (sot) are ...
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61 views

Show these sets are disjoint

This is a part of a much bigger theorem I'm trying to prove, but I'm stuck with a little finishing argument which is very reasonable (is it? I sure hope it's true tho) but I can't get it to be done. ...
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1answer
19 views

Show that exist subset U and V of Metric Space.

Let $(X,d)$ a metric space, let be $x_0 \in X$, such that $x_0 \notin A$, where A is a subset closed of $X$. Show that exist open subset U and V disjoints of $X$ , such that $x_0 \in U$ and $A \subset ...
3
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1answer
24 views

Is compact $T_1$ topological space hausdorff?

I'm in a middle of a very hard exercise which its goal is to prove that some space is hausdorff, but all I could show is that it is $T_1$. But I can also deduce that it is compact. Is that enough for ...
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3answers
46 views

Lebesgue Measure of $\mathbb{R} \times \{0\}$

I know this is probably obvious, and I know the answer is that it is (Lebesgue) measure zero, but I'm having a hard time wrapping my head around it. Looking for an intuitive explanation. Question: ...
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32 views

Let $A$ be the annulus in $\mathbb{C}$, what is the space $A/{\sim}$ generated by identifying all points on the “inner circle” with each other?

The annulus is $A=\{z\in\mathbb{C}:1\leq |z| \leq 2\}$ and the "inner circle" here is the set of points $\{z\in\mathbb{C}:|z|=1\}$. If we identify all points of the inner circle together, then, ...
3
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0answers
27 views

$d_1(x,y)=|x-y|$ & $d_2(x,y)=|\frac1x-\frac1y|$ inducing same topology on $[1,\infty)$?

My textbook says the following: Let $X=[1,\infty)$ and define $d_1(x,y)=|x-y|$ and $d_2(x,y)=|\frac1x-\frac1y|$ . Then $d_1$ and $d_2$ are inducing the same topology, $(X,d_1)$ is complete but ...
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1answer
29 views

Accumulation point is a limit of some sequences in first-countable space with Axiom of Choice.

I have a statement which says that Let $X$ be a first-countable space and $E$ be a subset of $X$. If $p$ is an accumulation point of $E$, then there exists a sequence in $E$ that converges to ...
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0answers
25 views

Convergence in Fréchet spaces and the topology

actually i have two questions concerning Fréchet spaces (i am not familiar with these spaces so i need some help). I have a vector space $V$ with a distance $$d(x,y) = \sum_{n\in \mathbb{N}} ...
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3answers
54 views

How many elements does the free product $\mathbb{Z}/2\mathbb{Z}\ast\mathbb{Z}/2\mathbb{Z}$ have?

Taken from Hatcher, I think the free product $G =\mathbb{Z}/2\mathbb{Z}\ast\mathbb{Z}/2\mathbb{Z}$ should have infinite elements taking the form of words containing alternate elements, i.e. $a, b, ab, ...
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1answer
15 views

Finite complement topology over $\Bbb{R}$ is not second-countable under ZF?

Under $\mathsf{ZF+AC_\omega}$, the space $\Bbb{R}$ with finite complement topology is not second-countable, since a countable union of countable subsets of $\Bbb{R}$. However, such argument doesn't ...
2
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1answer
29 views

Totally disconnected space that is not $T_2$

The Wikipedia article on totally disconnected spaces seems to imply they are not necessarily Hausdorff (they are all $T_1$ though). What's an example of a totally disconnected non $T_2$ space? (A ...
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1answer
14 views

Free action on space implies that each point has a neighborhood that has an empty intersection with translations

Suppose $G$ is a topological group, $X$ a topological space and $G \times X \rightarrow X$ group action that is continuous. Further, suppose that the action is free ($G_x = \{e\}$, for all $x$). What ...
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41 views

Could someone please explain these two theorems from Topology? [on hold]

I'm trying to understand the concept of box and product topology..and these two theorems confuse me. In the first one, I didn't understand from the part "since inv(proj_b(V_b)) is open in Prod(X_a)" ...
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1answer
50 views

What is the interior of a single point in a metric space?

Let $(X,d)$ be a metric space. We know that if $x \in X$ , then $Cl(\{x\})=\{x\}$, which implies that $\{x\}$ is closed. However if that's the case, what would the interior of $\{x\}$ be? I was ...
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40 views

A compact $n-1$ dimesional manifold embedded in $\mathbb{R}^n$ has no boundary

Under what conditions is it true that a compact $n-1$ dimesional manifold embedded in $\mathbb{R}^n$ has no boundary (or more generally, when a manifold is embedded in some topological space)? For ...
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0answers
28 views

Marking Integers Using a Wheel

Suppose I had a wheel of diameter one meter and I was charged with marking every meter along an infinite stretch of a beach. The strategy is to insert pegs into the wheel so that every point that is a ...
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1answer
27 views

Prove that set of isolated points in $X$ is dense in $X$

Let $A=\{\text{isolated points of } X\}$. $X$ is a countable complete metric space. Show that $A$ is dense in $X$. My attempt: Basically we want to show that $\bar A = X$. First, we show that ...
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1answer
52 views

A miscellanea of properties of the rational sequence topology on $\mathbb{R}$

For each $x\in \mathbb{R}-\mathbb{Q}$ fix a sequence of rational numbers $(y_i(x))_{i\in \mathbb{N}}$ which converges to $x$. For each irrational point $x$ and each $n \geq 1$ let $M_n(x) = \{x\}\cup ...
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0answers
14 views

Let $X$ be a normal space then there exists a continuous map $f : X → [0, 1]$ such that $f^{−1} (0) = A$ and $f^{−1} (1) = B$ [duplicate]

Let $X$ be a normal space with the property that every closed set in $X$ is a countable intersection of open sets in $X$. Then show that given $A, B ⊂ X$ closed, $∃$ a continuous map $f : X → [0, 1]$ ...
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2answers
24 views

Difference between Path, Curve, Graph and Trace

I am having difficulties in understanding the differences between these concepts. We have a new lecturer who loves writing down things in dense mathematical notation (I don't think that's bad but I am ...
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1answer
26 views

About connected topological subgroup

I'm trying to understand a proof of a theorem but I didn't understand a point. Let $G$ be an locally compact abelian group. Denote $G_0$ the connected component of $0$ (the identity of $G$). It's an ...
4
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1answer
45 views

Quotient topology by identifying the boundary of a circle as one point

The following is an example taken from Munkres topology book: I don't understand why does $X^{*}$is homeomorphic to $S^{2}$, is this a basic fact that I don't understand or is it an example of ...
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0answers
18 views

Order topology is regular and not normal

π-Base shows that linear order topology is not normal. But I remember in class the prof said order topology is normal. If $X$ is a set with linear order $<$, define a topology on X by letting ...
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2answers
34 views

$\{\infty\}$ open in $\mathbb N\cup\{\infty\}$ with $d(a,b)=|\arctan a-\arctan b|$?

Let $X=\mathbb N\cup\{+\infty\}$. I want to find two metrices inducing different topologies. Let $d_1$ be the discrete metric then all subsets of $X$ are open. (in particular $\{+\infty\}$) But now ...
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1answer
23 views

pointwise convergence of a filter on $\mathbb{R}^\mathbb{R}$

In my topology lecture we have defined pointwise convergence for filters on function spaces, say $\mathbb{R}^\mathbb{R}$. A filter $\varphi$ on $\mathbb{R}^\mathbb{R}$ converges pointwise to ...
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0answers
14 views

Is the set of non-degenerate symmetric matrices with signature (p,q) simply connected?

Let $M=\big\{A\in\text{GL}(n,\mathbb{R})|\;A^T=A\;,\; A \text{ has signature }(p,q)\big\}\;$ denote the set of real non-degenerate symmetric matrices with signature $(p,q)$, where $p$ is the number ...
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2answers
49 views

Every Cauchy Sequence in the real number line converges

Prove that every cauchy sequence in $\mathbb{R}$ converges proof: Let ($a_n$) where $n\in \mathbb{N}$ be a Cauchy sequence. Let's first prove that it is bounded. Choose $\epsilon = 1$, then for some ...
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0answers
49 views

homeomorphism between zero-dimensional Hausdorff and two-point space

Given $\left\{g_{\alpha}: \alpha\in T\right\}$ consists of all continuous functions from $A$ to $\{0,1\}$ ($A$ is a zero-dimensional Hausdorff space). Let $G =\prod_{\alpha\in T} g_{\alpha}: ...
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1answer
25 views

How is the boundary in product spaces defined?

The general question: how is the boundary defined in product spaces? Given two topological spaces $X,Y$, I'd say that $\partial(X\times Y)=\partial X\times\partial Y$. But looking at what follows it ...
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2answers
34 views

Identifying the two-hole torus with an octagon

I am aware that the 2-hole torus can be identified with the octagon with the equivalence relation as given in this picture: ...
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3answers
29 views

Show that the set of isolated points of $S$ is countable

Let $S$ be a subset of $\mathbb{R}^n$; show that the set $I$ of isolated points of $S$ is countable. Let $\mathbf{x}\in I$. There exists an open ball, say $B(\mathbf{x},r_\mathbf{x})$, of radius ...
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1answer
13 views

Application of Urysohn's Lemma to non-disjoint closed sets

Let $X$ be a normal space with the property that every closed set in $X$ is a countable intersection of open sets in $X$. Then show that: (a) Given $A \subset X$ closed, $\exists$ a continuous map ...
2
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1answer
21 views

“Closed interval” on ordered topology

Let $X$ be linearly ordered by a relation $\leq$. Taking as a subbase for topology on $X$ all sets of the form $\{x;x<a\}$ and $\{x;x>a\}$, for $a\in X$. Can be $\{x\in X; a\leq x\leq b\}$ a ...
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1answer
27 views

Is the saturation of Borel sets Borel?

Problem. Let $G\times X\rightarrow X$ be a continuous action of a Polish group on a Polish space. Let $A\subseteq X$ be Borel. Is the saturation $[A]_{G}:=G\cdot A$ a Borel set? One approach. The ...
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1answer
41 views

Let $X$ be a normal space then there exists a continuous map $f : X → [0, 1]$ such that $f^{−1} (0) = A$

Let $X$ be a normal space with the property that every closed set in $X$ is a countable intersection of open sets in $X$. Then show that given $A \subset X$ closed, there exists a continuous map $f : ...
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0answers
24 views

Precompact and locally finite implies finite intersection

An exercise in Lee's Introduction to Smooth Manifolds asks the following: Let $M$ be a topological manifold, and let $\mathcal U$ be an open cover. Suppose the sets in $\mathcal U$ are precompact ...
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1answer
19 views

Give a example of a sequence of continuous functions which do not form a Cauchy sequence

As an example that not every Cauchy sequence in $(M,d)$ is converging in $M$ the following examples are given: Consider $(\mathbb{Q},d_{\text{eucl}})$ and a sequence $q_n \in \mathbb{Q}\to ...
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1answer
21 views

Is Alexandroff duplicate compact?

Consider the Alexandroff duplicate $X\times_{ad} 2$, the space $X\times 2$ where the points of the form $(x,1)$ are isolated and for each open set $U$ in $X$, $(U\times\{0,1\})\setminus (x,1)$ is ...
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1answer
37 views

Find a quotient map $f:(0,1) \rightarrow [0,1]$ where the intervals $(0,1)$ and $[0,1]$ are in $\mathbb{R}$ and endowed with the subspace topology.

Find a quotient map $f:(0,1) \rightarrow [0,1]$ where the intervals $(0,1)$ and $[0,1]$ are in $\mathbb{R}$ and endowed with the subspace topology. I am really not to sure where to start. I know ...
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1answer
71 views

Finding a homeomorphism between $ B $ and $ B \times B $.

Let $ B = A^{\mathbb{N}} $, where $ A $ is a topological space. Show that $ B $ is homeomorphic to $ B \times B $. My failed attempt: I’m trying to show that there exists a function $ f: B \to B ...
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2answers
72 views

Is any compact, path-connected subset of $\mathbb{R}^n$ the continuous image of $[0,1]$?

If $f:[0,1] \to \mathbb{R}^n$ is any continuous map, then the image $f([0,1])$ is a compact, path-connected set, which is easy to show using some elementary topology. My question is the converse: ...
2
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1answer
77 views

Showing a function $f:X \rightarrow Y$ is continuous

I am working through some practice questions, and I am not sure if I am on the right track with this one: Let $X = \cup_{n≥1}A_n$, be a topological space and assume that a map f : X → Y is such ...
4
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2answers
56 views

Countable dense subsets of $\mathbb R$ are homeomorphic

Suppose countable subsets $A,B$ of the real line $\mathbb R$ satisfy $\overline{A}=\overline{B}=\Bbb R$. How can one show that $A$ is homeomorphic to $B$? I even have no idea how to get a bijection ...