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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24 views

Find a circle which is a strong deformation retract of $\mathbb{R^2}-x_0$

Let $x_o\in\mathbb{R^2}$. Find a circle which is a strong deformation retract of $\mathbb{R^2}-x_0$ Proof: If $x_0=(a,b)$, Let $S=\{(x-a)^2+(y-b)^2=1\}$. Then $f_t(r,\theta) = (\exp((1-t)r),\theta)$, ...
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1answer
20 views

Prove that $h$ and $p*q$ are homotopic relative to {$0,1$}

Let $0<s<1$. Given paths $p$ and $q$ with $p(1)=q(0)$, define $h$ by the formula $$h(t) = \begin{cases} p(t/s),& \text{if} \quad 0 \leq t \leq s \\ q((t-s)/(1-s)), &\text{if} \quad ...
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2answers
33 views

Union of a sequence of path connected sets being path connected

Theorem: Let ${A_n}$ be a sequence of path connected subsets of a space $X$ such that for each integer $n\ge 1$, $A_n$ has at least one point in common with one the preceding sets $A_1, ..., A_{n-1}$. ...
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80 views

Prove that the Pontryagin dual of a locally compact abelian group is also a locally compact abelian group.

Let $ G $ be a locally compact abelian (LCA) group and $ \widehat{G} $ the Pontryagin dual of $ G $, i.e., the set of all continuous homomorphisms $ G \to \mathbb{R} / \mathbb{Z} $. Clearly, $ ...
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0answers
34 views

Exercise in Section 2.4 of Singer & Thorpe

I'm trying to solve the exercise in Section 2.4 of Singer & Thorpe, which is to prove that if $S$ is a compact Hausdorff topological space and $(U_n)_{n \in \Bbb N}$ be a family of dense open ...
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2answers
142 views

Compact metric connected space

If I have a compact metric space $X$ such that for all $a,b \in X$, there are points $a:=x_1,...x_n=:b$ such that $d(x_i,x_{i+1})< \varepsilon$, then this space is connected. Somehow, I don't see ...
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17 views

Showing that the function $C(b)$ is a compact set for $|b| < 1$

I am reading "An Invitation to Dynamical Systems", and one of the challenge problems is to prove that $C(b)$ is a compact set where $C(b)$ is defined as the set of all numbers that can be expressed in ...
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1answer
59 views

Hausdorff dimension mathces Box-counting dimension

I need to compute the Hausdorff dimension of certain sets using a computer and, to date, my approach has been to use a Box-counting algorithm, for I once read that the Hausdorff dimension of an ...
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3answers
50 views

Compactness, topology

In a general topological space $(X,\tau)$ I have the following situation: $$F\subset M\subset N$$. If I prove that $F$ is compact in $N$ (w.r.t the induced topology), is it true that $F$ is compact ...
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1answer
29 views

A space $X$ is path connected if and only if there is a point $a$ in $X$ such that each point of $X$ can be joined to $a$ by a path in $X$.

A space $X$ is path connected if and only if there is a point $a$ in $X$ such that each point of $X$ can be joined to $a$ by a path in $X$. I'm trying to prove this statement. The only if direction ...
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1answer
51 views

Show that a sequence of functions is not compact in $(C^1([-1,1]),\lVert\,\cdot\,\rVert_{\infty})$

Let $F:=(f_n)_{n\in\mathbb{N}}$ where $$\forall x\in[-1,1]\,\,\,\,\,\,\,\,\,f_n(x):=\mid x\mid^{1+\frac{1}{n}}$$ I have to prove that $F$ is not compact in ...
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0answers
18 views

Uniform convergence of $f_n(z)=\sum_{j=o}^n z^j$ on the open unit complex disk.

I have the sequence $f_n(z)=\sum_{j=o}^n z^j$ on the open unit complex disk ($\Delta$). My question is whether or not my approach is correct to the following problems: is the sequence normal? Does ...
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1answer
17 views

Prove that intervals of the form $(a,b]$, $[a,b)$, $(-\infty,a]$, $[a,\infty)$ do not have the fixed point property.

Prove that intervals of the form $(a,b]$, $[a,b)$, $(-\infty,a]$, $[a,\infty)$ do not have the fixed point property. In the case of open intervals, I can derive that they do not have the fixed point ...
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68 views

Prove that an open interval and a closed interval are not homeomorphic

Prove that an open interval $(a,b)$ and a closed interval $[c,d]$ are not homeomorphic. I'm trying to prove this statement but have only vague ideas on how to start. How may I use the property of ...
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1answer
24 views

G is open in X iff $\overline{G \cap \bar{A}}=\overline{G \cap A}$ for all $A\subset X$

It is clear that $\overline{G \cap {A}}\subset \overline{G \cap \bar{A}}$ but not getting how to show the reverse inclusion!
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1answer
27 views

Algebraic rules for a fundamental polygon

If we have a 2d surface, we can give it a plane model with a sequence of letters corresponding to gluing instructions for it. http://en.wikipedia.org/wiki/Fundamental_polygon for examples. So if we ...
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1answer
37 views

Assigning manifolds to graphs in a functorial way

I am looking for ways to functorially assign manifolds (or more general topological spaces) to families of graphs. To be more precise, I am interested in functors from specific subcategories of the ...
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1answer
57 views

Closed and Compactness on $\mathbb Q$ (Multiple Choice)

Please help me regarding the following question. Consider $\mathbb Q$ with usual metric (i.e $d(p,q)=|p-q|$).Then which of the following are true? $\{q\in\mathbb Q|2<q^2<3\}$ is closed ...
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53 views

How can I envision the open ball around the french railways metric?

I have that the French railways metric defined by a metric space $(\mathbb{R}^2,d)$ has a distance function that is a metric as follows: $$d(x,y) = \begin{cases} \|x-y\|, & \text{if $x,y,0$ are ...
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3answers
29 views

basic question about open balls

Let $U$ be an open set in a complete normed space. Let $x \in U$. Hence, we can find an open ball $B$ centered at $x$ that lies inside $U$. Question: Does it follow that $U - x $ contains an open ...
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1answer
40 views

$\{{(x,K) : x \in K}\}$ is closed in $X \times \{\text{closed subsets of }X\}$

Let $X$ be a locally compact (and Hausdorff, if needed) space. Let's denote the closed subsets of $X$ by $C(X)$, equipped with the Hausdorff topology. I want to prove that $F=\{{(x,K) : x \in K , K ...
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1answer
16 views

Dilation of a set is a closed set?

Let $X$ be complete normed space. Suppose $B \subseteq X$ is a set. Define for scalar $k$ $$ kB = \{ x \in X : x = kb , \; \; \; b \in B \} $$ Is $kB$ a closed set ?
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3answers
30 views

If $A_1\cap…\cap A_n \neq \emptyset$, does $(A_1\cap…\cap A_n)^{c} =A_1^{c} \cup … \cup A_n^{c} = \emptyset$?

If I have some collection of sets such that $A_1\cap...\cap A_n \neq \emptyset$, then what happens if I apply the complement (denoted by superscript c) to both sides? i.e., $(A_1\cap...\cap A_n)^{c} ...
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1answer
54 views

Is Alexandroff Duplicate A(X) of X paracompact?

Prove or disprove: If $X$ is a paracompact space, then Alexandroff Duplicate $A(X)$ of $X$ is paracompact. Thanks for any help. ...
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28 views

Is the closure of the intersection of a set with a closed subspace is equal to the closure of the set intersection the subspace?

If $M$ is a closed subspace of a topological vector space $X$ and $A$ intersects $M$. Is $\overline{A\cap M}=\overline{A}\cap M$
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1answer
39 views

The topology defined by the family of pseudo-distances.

A pseudometric (aka. pseudo-distance) is a metric except that maybe $x \neq y$ but $d(x,y) = 0$. Consider a family $(d_a)_{a \in A}$ of pseudometrics on a set $E$. For each $x \in E$ and each finite ...
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40 views

Showing that an identity map for a metric space in $\mathbb{R}^2$ is continuous but that its inverse isn't.

Suppose that $A=(\mathbb{R}^2,d)$ is a metric space with $d(x,y)=||x-y||$. I would like to show that if I have an identity map from $I:A \to \mathbb{R}^2$ with its euclidean distance function, then ...
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93 views

Is every compact space compactly generated?

I am using the definition of compactly generated space from The Category of CGWH Spaces, which is In $\mathcal{Top}$, $k$-closed subset $Y\subset X$, means $u^{-1}(Y)$ is closed in $C$ for any $u: ...
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3answers
34 views

Proving connectedness

Suppose $S$ is a set, any pair of points of $S$ ($P,Q$ assume) can be contained in a connected subset of $S$. Show $S$ is also connected. I tried to use the polygonal chain theorem(Every open set ...
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1answer
23 views

Openess of sets given by equivalence relations in the quotient topology.

I'm trying to prove this: Let $R$ be an equivalence relation in $X$. Show that $A$ is open in $X/R .\iff \bigcup_{[x]\in A}[x]$ is open in $X$ One of the first things that come to my mind is ...
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1answer
24 views

Define $F:S^2\longrightarrow \mathbb R$ continuous such that $F|_{H_+}=f$ and $F|_{H_-}=g$?

Let $f, g:D^2\longrightarrow \mathbb R$ be continuous such that $f|_{\partial D^2}=g|_{\partial D^2}$. I want to show there exists $F:S^2\longrightarrow \mathbb R$ continuous such that $F|_{H_{+}}=f$ ...
2
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1answer
39 views

Why is there a bijection between the ultrafilters that converge and a topology

If we call $\mathcal{UF}(X)$ the set of ultrafilters on a set $X$. I read here that there is a bijection between topologies on a set $X$ and $\{0,1\}^{\mathcal{UF}(X)}$. As I am unfamiliar with ...
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3answers
57 views

What is a “control point”?

I'm trying to figure out a good definition of control point for use in wikipedia (see https://en.wikipedia.org/wiki/Control_point_(mathematics) ) There seems to be a bias towards ascribing a ...
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1answer
51 views

Projection onto a convex closed set

H, If $K$ is a non-empty convex and closed subset of a uniformly convex Banach space $X$ (Hilbert for example) and $v \notin K$, we know that there exists a unique $k_0\in K$ such that ...
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2answers
64 views

Prove that a set is a topology. [closed]

So I know the definition of topology but I find it really difficult to know what to do to prove a given set is a topology. I was wondering if someone could talk me through a few problems. ...
3
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3answers
180 views

How to explain why free loop space, or based loop space, is infinite dimensional to non-math people.

I am giving a math talk to non-mathematicians. I was wondering how to explain how the free loop space, or based loop space, of a topological space is infinite dimensional so that a non-mathematician ...
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1answer
23 views

Approximation of a Function Discontinuous only at a Set of First Category

Consider a function $f$:$T$ $\rightarrow$ $R$ that is continuous except for a set of first category. Brosowski and da Silva (1997) argue that there exist a sequence of continuous functions that ...
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2answers
47 views

2D grid which is topologically equivalent to a sphere?

I admit that my knowledge of topology is limited to the idea that a mug and doughnut are homomorphic since you can morph one into the other with a continuous deformation. I am a game dev working on a ...
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1answer
62 views

Reflexive spaces of $\ell_{\infty}$

Let $\ell_{1}$ and $\ell_{\infty}$ the usual spaces of the convergente and bounded sequences, with its usual norms. As we know, the topolical dual of $\ell_{1}$ is $\ell_{\infty}$, that is ...
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1answer
51 views

Completeness of the metric r(x,y)=min{d(x,y},1}

Can somebody help me out with the following: If (X,d) is a metric space and r(x,y)=min{d(x,y},1} for all x,y in X. I proved that r is a metric and that r and d are equivalent. Now I want to prove ...
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1answer
37 views

Is image of every closed set is closed set ?

If f is continuous and bijection function of Hausdorff space into topological space. Is image of every closed set is closed set ? Thanks.
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2answers
72 views

Proving a metric induces the product topology

Let $(M,d)$ and $(N,d')$ be metric spaces. Prove that the product topology is induced by the metric $d_1((x,y),(x',y')=d(x,x')+d(y,y')$ and ...
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1answer
33 views

Inclusions in Metric Topology

Prove that if $\exists L>0 : d_2(x,y)\geq Ld_1(x,y)$ then the topology induced by $d_2$ is finer than the topology induced by $d_1$. I'll call $T_1$ the topology induced by $d_1$ and $T_2$ the ...
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1answer
31 views

Are these definitions equivalent for a subbasis of the product space?

I'm trying to determine how the definition of the product topology for $Y^X$ I found in some notes from math.bard.edu relates is the same as the product topology defined by Munkres. One of the things ...
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2answers
84 views

Examples of Stone algebras which are not Boolean algebras

Grätzer, in his Lattice Theory: Foundation, describes a Stone algebra as a distributive lattice with pseudocomplementation $L$ which satisfies the Stone identity: for every $a \in L$, $\neg a \vee ...
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21 views

Neighborhoods for continuous functions between CG spaces

I have a couple of problems regarding the existence of certain neighborhoods, so as to prove continuity of suitable functions. Suppose then that $Y,X$ and $Z$ are compactly-generated Hausdorff spaces ...
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2answers
67 views

Prove that a quotient of an n-dimensional ball by an equivalence relation is homeomorphic to an n-dimensional sphere.

Question Prove that a quotient of an n-dimensional ball by an equivalence relation, whose only non-trivial equivalence class is the n-1 dimensional sphere, is homeomorphic to an n-dimensional sphere. ...
2
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3answers
56 views

Which sequences converge in the topological space $\left(\mathbb{N}\times\mathbb{N}, \mathcal{J}_{\text{Lexicographic}}\right)$?

Which sequences converges in the topological space $\left(\mathbb{N}\times\mathbb{N}, \mathcal{J}_{\text{Lexicographic}}\right)$? That i have tried: For each $(p,q)\in \mathbb{N}\times\mathbb{N}$ we ...
1
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1answer
48 views

discontinuity point of a monotone function of $R^n$

I am aware that a monotone function from $R$ to $R$ has an at most countable points of discontinuity. Can we say that the set of points of discontinuity of a monotone function from $R^n$ to $R$ ...
2
votes
1answer
66 views

Find a surjective function $f:B_n \rightarrow S^n$ such that $f(x)=f(y) \iff \|x\|=\|y\|$

Let $B_n = \{x \in \mathbb{R^n} : \|x\| \le 1\}$ and $S^n = \{x \in \mathbb{R^{n+1}} : \|x\| = 1\}$. Find a surjective function $f:B_n \rightarrow S^n$ such that $f(x)=f(y) \iff \|x\|=\|y\|$. ...