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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48 views

Prob. 6, Sec. 20 in Munkres' TOPOLOGY, 2nd ed: How is this set not open?

Let $\mathbb{R}^\omega$ denote the set of all sequences of real numbers, let $\tilde{\rho}$ denote the uniform metric on $\mathbb{R}^\omega$ defined as $$ \begin{align*} \tilde{\rho}(x,y) \colon= ...
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49 views

Product Spaces: Tube Lemma

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. My professor asked to prove the following Lemma. The Tube Lemma: Let $K$ be a compact ...
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1answer
31 views

Let $X\subset\mathbb{R}^3$ be the union of the coordinate axies, I want to show that $\mathbb{R}^3-X$ is homotopy equivalent to a graph

Let $X\subset\mathbb{R}^3$ be the union of the coordinate axies, I want to show that $\mathbb{R}^3-X$ is homotopy equivalent to a graph, and the question asks further "which graph" Let ...
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0answers
52 views

Hilbert Space is not locally compact.

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. Show that Hilbert Space is not locally compact at any point. This is what I understand: ...
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2answers
58 views

Problem of General topology (Schaum)

Let $\tau=\{U \subseteq \mathbb{Z}_+ :$ if $n \in U \implies$ all divisor of $n $ is in $U \}$. Let $f: (\mathbb{Z}_+,\tau) \to (\mathbb{Z}_+,\tau)$. Prove: $f$ is continuous $\leftrightarrow$ if ...
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1answer
37 views

Closed set in Baire space

I am reading a book on Banach spaces. It introduces the Baire space $\mathcal{N}=\mathbb{N}^\mathbb{N}$ as the product of infinitely many copies of $\mathbb{N}$ with the discrete topology. We have ...
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0answers
47 views

Uncountability of $\mathbb{R}^I$ if $I$ is uncountable

Prove that if $I$ is uncountable, then $\mathbb{R}^I$ with the product topology is not countable. Based on what I have read, a set is uncountable if there is a bijection from that set to ...
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1answer
21 views

Is the co-limit of a chain of normal subspaces necessarily normal?

Suppose $ X_0 \subset X_1 \subset X_2 \subset \dots$ is a chain of normal subspaces of $X$ such that $X= \cup_{i=1}^{\infty} X_i$. Assume that $X$ has the colimit topology w.r.t. these subspaces. Can ...
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29 views

Prove $X \setminus \operatorname{Cl}A = \operatorname{Int}(X \setminus A)$

Definitions ($X$ is a topological space): • $\operatorname{Cl}A$ is the intersection of all closed subsets of $X$ which contain $A$ as a subset. • $\operatorname{Int}A$ is the set of all ...
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31 views

Simple homotopy construction

I'm sure this isn't too difficult but i can't seem to do it if you have two loops $p_0 = e*g $ and $p_1 = g*e$ where $e$ is the trivial loop How would i construct an explicit homotopy between the ...
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31 views

Is the following a valid characterisation of complete metric spaces?

A metric space $(X, d)$ is called complete if and only if every Cauchy sequence converges. Now does the following hold: A metric space is complete if and only if every sequence $(x_i)_{i\in\mathbb ...
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20 views

Closure of $\{ ( e^t\cos t,e^t\sin t) : t \in \Bbb R \}$

Suppose $\alpha: \Bbb R\to \Bbb R^2$ given by $\alpha (t)=(e^t \cos t,e^t \sin t)$, $A=\alpha(t)$ is a smooth manifold. What is the closure of $A$? I know that the closure of the set is the set ...
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39 views

$\mathbb{Q}$ with topology from $\mathbb{R}$ is not locally compact, but all discrete spaces are

Wikipedia claims that all discrete topological spaces are locally compact but that $\mathbb{Q}$ isn't when endowed with the topology of $\mathbb{R}$. I don't know if I understand those examples ...
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33 views

Weak topology and the closed unit ball

I want to prove that there is no neighbourhood of $0$ in the closed unit ball. I can use pointwise and Banach-Alaoglu theorems to prove it.
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44 views

Tietze Extension Theorem ,,

If we have X a normal space, C a closed subspace of X, Y a completely regular space, and $f:C \rightarrow Y$ a continuous function. How do we show that f has a continuous extension $F: X \rightarrow ...
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1answer
58 views

Spaces in which “$A \cap K$ is closed for all compact $K$” implies “$A$ is closed.”

Let $X$ denote a topological space. For any $A \subseteq X$, consider two possible conditions on $A$. $A$ is closed $A \cap K$ is closed, for all compact $K \subseteq X$. If $X$ is Hausdorff, then ...
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1answer
31 views

if $V_1\cong U_1, V_2\cong U_2$, is $(V_1\cup V_2 \cong U_1\cup U_2)$? Pasting homeomorphisms

My question arises from the theory of covering spaces. assume $f:Y\to X$ is a covering map, or more generally a local homeomorphism. Assum $U_1,U_2\subset X$ are open sets such that $f|_{V_1}, ...
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42 views

Prove that this infinite sum involving metrics is also a metric

The following is a question from a previous assignment that I was unable to complete. Any assistance on how to complete this would be appreciated. Let $\varrho_i: X\times X\to \Bbb R^+$ with ...
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2answers
65 views

Prove that this is a metric space

The following is a question from a previous assignment that I was unable to complete. Any assistance on how to complete this would be appreciated. Let $\varrho: X\times X\to \Bbb R^+$ be a metric on ...
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97 views

Open Unit Ball diffeomorphic to the Open Unit Cube

How can I show that the open unit cube $(-1,1)^n \subset \mathbb{R}^n$ and the open unit ball $B = \{x \in \mathbb{R}^n \mid \|x\| < 1\}$ are diffeomorphic? I know that one can proof this by ...
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67 views

Proving the continuity of functions from metrics

Context: I'm studying mathematics at university, and am having trouble with some of the continuity questions. The following is a question from a previous assignment that I was unable to complete. The ...
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1answer
35 views

Embeddability of connected sum of non-embeddable surfaces

Let $X$ be a surface which can not be embedded into $\Bbb R^n$. Let $X \# X $ denotes the connected sum of two copies of $X$. Then is it true that the connected sum $ X \# X $ is also not embeddable ...
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49 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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56 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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31 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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72 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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12 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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49 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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88 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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33 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 ...
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30 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
81 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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40 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, ...
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35 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
44 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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1answer
39 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
76 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
28 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 ...
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1answer
34 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
22 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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1answer
66 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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38 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
56 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
68 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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2answers
66 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
33 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 ...
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1answer
68 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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33 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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40 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
65 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 ...