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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10 views

Describing the clopen sets of a profinite group

I've read somewhere that all clopen subsets of a profinite group $$G \simeq \underset{\longleftarrow}{\operatorname{lim}} \left(G_i, f_{ij}:G_i \to G_j\right)_{i,j \in I}$$ are exactly the preimages ...
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1answer
15 views

Spaces homotopy equivalent to finite CW complexes

I'm doing a project about Topological Complexity (it doesn't matter what it is for the questions I will ask) and I have proofs for a few results about the bounds of the topological complexity of ...
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1answer
27 views

Prove closed disc $D^n$ is homeomorphic to the cone $CS^{n-1}$

I need to find a continuous surjective map from $D^n$ to $CS^{n-1}$. For 2 dimensions, we can use $$f: S^1 \times I /S^{1} \times \{1\} \rightarrow D^2$$ with $f(\theta,t) = (1-t)e^{i \theta}$ ...
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30 views

Does a map between topologies determine a map between sets?

Let $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ be Hausdorff spaces. Consider a function \begin{equation*} \phi:\mathcal{B}\rightarrow \mathcal{A} \end{equation*} which preserves inclusion, arbitrary ...
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1answer
23 views

Is torus w. disc removed homotopic to klein bottle w. disc removed?

I know that homeomorhic spaces are homotopic, but am not sure if this applies, since I think they are not homeomorphic due to orientability. I know f and g are homotopic if they represent: ...
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1answer
55 views

The magic of the morphisms

Given a set $X$. Let $S\subseteq X$ and consider $(X,S)$ as a very simple mathematical structure, lets call it a spotted set. Given two spotted sets, then a morphism $\alpha ...
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1answer
18 views

Differentiability of norm

Is a norm from any normed space $\mathbb{A}$ to $\mathbb{R}$ Frechet differentiable at zero? I've tried proving by contradiction that the norm is not Frechet differentiable at zero, but didn't get ...
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1answer
52 views

Showing that $\displaystyle\underset{n\rightarrow \infty}{\lim}\int_0^1 f_n = \int_0^1\underset{n\rightarrow \infty}{\lim} f_n$

How to solve the following task: Show that if $f_n$ is a sequence of uniformly converging mappings $f_n \in C[0,1]$, where $C[0,1]=\{f:[0,1]\rightarrow\mathbb{R} \;\mid\; f\; ...
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0answers
18 views

any sum of sets open\nullset is a set of the same form

I'm curious how can one prove that any sum of sets $G\setminus N$, where $G$ is open and the Lebesgue measure of $N$ is 0, is a set of the same form. it is easy for countable sums, but in general? ...
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9 views

Holonomic splitting

I am reading the book "Introduction to the h-Principle" by Eliashberg and Mishachev. At the moment I try to understand the Section 1.7 Holonomic splitting on page 12 but without success. I do not ...
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2answers
31 views

Conditions for a homeomorphism

In my topology notes the definition is given as: A function $f : X \rightarrow Y$ is said to be a homeomorphism if: $f$ is continuous, bijective, and moreover its inverse $f^{−1} : Y \rightarrow X$ ...
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26 views

how can i prove that 3 problems [on hold]

Prove that any subspace of a discrete space is discrete. Prove that any subspace of an indiscrete space is indiscrete. Prove that if A C X is r-open, then any r A -open set is also r-open.
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1answer
29 views

Fundamental Property of Regular CW Complexes

$\newcommand{\R}{\mathbf R}$ For a cell $e$ in a CW complex, we write $\partial e$ to denote $\bar e-e$. Note that $\partial e$ may not be the topological boundary of $e$ in $X$. A CW complex ...
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1answer
11 views

Equivalent properties for a kind of relative compactness?

Let $X$ be a topological space and $Y \subseteq X$. Consider the following statements: (i) Every net in $Y$ has a cluster point in $X$. (ii) Every infinite subset of $Y$ has a complete accumulation ...
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1answer
14 views

Hybrid equivalence of Polynomial-like maps

I am reading Douady and Hubbards "On the dynamics of polynomial-like mappings". I am relatively new dynamics of complex maps, and I would appreciate some help with aspects of the following. ...
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1answer
17 views

Should a compact nbd of $p$ contain some open nbd containing $p$?

If some topological space $X$ is locally compact, then each point of $X$ has at least one compact neighborhood. The book I'm reading now doesn't mention if such compact neighborhood of a point should ...
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1answer
35 views

Proving $\mathbb{R}/\sim$ is homeomorphic to unit circle

Let $S$ be the unit circle in $\mathbb{C}$, standard topology. Define the equiv. rel. $\sim$ on $\mathbb{R}$ as $x\sim y\iff x - y\in\mathbb{Z}$. I would like to prove that $\mathbb{R}/\sim$ is ...
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1answer
19 views

The degree of a circle function

Does the degree of a circle function $f:S^1 \to S^1$ simply mean how many times the mapping of $f$ wraps around the $S^1$?
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2answers
14 views

Prove that the space $\Bbb R_K$ is not regular.

Prove that the space $\Bbb R_K$ is not regular. where the basic open sets on $\Bbb R_K$ is given by $\{(a,b):a,b\in \Bbb R\}\cup \{(a,b)-K\}$ where $K=\{\dfrac{1}{n}:n\in \Bbb Z_+\}$. ...
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2answers
48 views

If $x$ is an isolated point of $S \subseteq \mathbb{R}$, then $x$ is a boundary point of $S$. [duplicate]

Is the following proof valid? (Note: I know there is a post discussing this problem, but I am curious to see if my argument works). This problem is different from another post that is similar with ...
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0answers
33 views

Let $S$ be the set . Which of following are true?

Let $S =\{\frac{1}{3^m}+\frac{1}{7^n}$ , where $m,n \in \mathbb N\}$ Then A.$S$ is closed B.$S$ is not open C.$S$ is connected D.$0$ is a limit point of S I see that $0$ is limit point of $S$ but ...
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1answer
49 views

What are the two disjoint closed sets that cannot be separated by two disjoint open neighborhoods in the Ellentuck topology?

Denote by $X := [\mathbb{N}]^\infty$ the set of infinite subsets of $\mathbb{N}$. Recall that the Ellentuck topology is a topology on $X$ generated by sets of the form $\{A\text{ infinite} \mid ...
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1answer
25 views

Prove some identities involving interior, closure in topology

I want to prove the following statements: (i) $X \setminus Y ^ \circ $ = $cl ( X \setminus Y) $ I wrote down that $Y ^\circ$ is open so $Y ^\circ = Y $. Therefore $X \setminus Y ^ \circ $ is ...
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1answer
52 views

What kinds of transformations preserve network topology?

I have been reading a number of "network science" papers where the authors perform transformations on networks that seem to preserve the topology of those networks. By "topology", I mean a collection ...
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1answer
20 views

Connected sets definition [duplicate]

Defn: A set $X$ is connected if there do not exist non-empty, disjoint open sets $U,V$ s.t $U$ $\cup$ $V$ $=X$. I thought intuitively that this meant that this was like the English dictionary ...
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1answer
23 views

Compact sets in the product of topological spaces.

Let $G_1$ be a non-compact topological space and let $G_2$ be a generic topological space. What are the compact sets in the product $G_1\times G_2$? Surely we can take the sets of the form $K_1\times ...
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1answer
23 views

The point-wise closure of the space of continuous functions

Let $X$ be a locally compact space and consider $C_0(X)$. We denote $b(X)$, by the set of all bounded functions on $X$. It is easy to be checked that $b(X)$ may be considered as a C*-sub algebra of ...
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2answers
31 views

Interior, closure and boundary of subset in $\mathbb{R}$ \ $\mathbb{Q}$.

I have the subset $\left[0,1\right] \backslash \mathbb{Q}$ in $\mathbb{R} \backslash \mathbb{Q}$. Am I right in thinking that this set is open and not closed in the space given? Also, how do I go ...
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1answer
48 views

Injectivity in the zero homology

I'm struggling with following step in an excercises about Mayer-Vietoris sequences: In one step the solution says this map is injective since $A \cap B$ is path-connected: $$ H_0(A \cap B) ...
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1answer
71 views

Property of Nowhere Dense Sets

I am trying to prove the following statement regarding nowhere dense sets: "In a metric space X, the frontier of an open set is the set of accumulation points of a discrete set." As far as my ...
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2answers
27 views

Show that if $\prod_\alpha X_\alpha$ is normal then so is $X_\alpha$.

Show that if $\prod_\alpha X_\alpha$ is normal then so is $X_\alpha$. This a question of proof-verification.So please suggest the required edits and fault in the logic but please don't give a ...
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0answers
53 views

Find interior points, boundary points, cluster points, limit points and isolated points of a set

Determine the interior points, the boundary points, the cluster points, the limit points, and the isolated points of each of the following subsets of $\mathbb R^2$. Also, classify each of the sets as ...
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0answers
47 views

Homeomorphism between product of spheres and euclidean space

I need to prove that ${S^n} \times {S^k}$ is homeomorphic to a subspace of ${\mathbb{R}^{n + k + 1}}$ by constructing an explicit map between the two. I am unsure how to start this as I can't seem ...
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0answers
31 views

Describe set of $z^2$ as z moves over 2nd quadrant and show it is open and connected

Problem: Describe the set of points $z^2$ as $z$ varies over the second quadrant: {z = x + iy; x < 0 and y > 0}. Show this is an open connected set. (Hint: use the polar representation of z.) The ...
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4answers
54 views

for $I = [0,1]$, is $I\times I$ convex in $\mathbb{R} \times \mathbb{R}$?

for $I = [0,1]$, is $I \times I$ convex in $\mathbb{R} \times \mathbb{R}$? The definition of convex seems to be that $Y \subset X$ is convex in $X$ if $\forall a < b $ in $Y$ whole of $(a,b)$ ...
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0answers
32 views

What simple topological properties of conic sections can be explored?

In the framework of my science fair project I am working on conic sections in different metric spaces. What simple topological properties/operations and so can I explore on them? Edit: To clarify, ...
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3answers
54 views

A non continuous linear map $A:\Bbb{R}[X]\rightarrow \Bbb{R}$ such that $A(P)=P(1).$

I have a linear map $A:\Bbb{R}[X]\mapsto \Bbb{R}$ such that $A(P)=P(1)$, for the $p-$norm : $\Vert P\Vert=\bigl(\sum_{i=1}^n\vert a_i\vert^p\bigr)^{1/p}$ where $p\in[1,+\infty].$ For the cas ...
2
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1answer
23 views

Does a neighborhood of a point include that point?

I'm working through the topology problem set in baby Rudin and I'm wondering if a neighborhood $N_r(p)$ of a point $p$ in a metric space automatically contains $p$, or just $\forall q|d(p,q)<r$, ...
2
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1answer
38 views

Topological spaces that remain non-metrizable, if the definition of metric space allows $d(x,y) = 0$ where not necessarily $x = y$?

In the definition of metric space, only one thing strikes me as unnatural: the requirement that $d(x,y) = 0$ implies $x = y$. As a programmer, I don't find it uncommon to deal with equivalence ...
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2answers
28 views

Proof that a subset of metric space with euclidian norm is open iff the same subset is open in metric space with Manhattan norm

For $\mathbb{R}^2$ we have the euclidian norm $$(x_1,x_2)\mapsto\sqrt{x_1^2+x_2^2},$$ and the Manhattan norm $$(x_1,x_2)\mapsto|x_1|+|x_2|.$$ Let $d_E$ and $d_M$ be the metrics defined by these norms, ...
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1answer
21 views

Connected Components of p-adic rationals

Notation: $p$ - a prime integer, $\Bbb{Z}_p$ - set of $p$-adic integers, $\Bbb{Q}_p$ - set of $p$-adic rationals, $\Bbb{Q}$ - set of rationals, $\Bbb{R}$ - set of reals. While reading up on ...
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1answer
49 views

Open or closed set in $\mathbb{R}$

I have this set $A=\left\{\frac{1}{n}|n\in\mathbb{N}\right\}$ I need to show that it is neither open or closed in $\mathbb{R}$. And that the union ...
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1answer
36 views

Is there a $T_6$ space in which a sequentially (or countably) compact subset is not closed

It is known that a $T_2$ space $X$ is $KC$, i.e. every compact subset of $X$ is closed. The space $[0, \omega_1]$ is $T_5$ but not $T_6$ and the subset $[0, \omega_1)$ is sequentially compact (and ...
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1answer
37 views

The space of continuous functions on an interval has a countable dense subset and a countable basis

Give $\Bbb R^I$ the uniform metric, where $I = [0, 1]$. Let $C(I, \Bbb R)$ be the subspace consisting of continuous functions. Show that $C(I, \Bbb R)$ has a countable dense subset, and therefore a ...
2
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1answer
26 views

A subspace of a mapping space?

We have a set $$ M=\{f:\mathbb{R} \rightarrow \mathbb{R}\mid f(1)>0\}\;.$$ I have never encountered this kind of set before. I assume it is correct to say that $M$ is a subspace of a mapping ...
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2answers
56 views

Familiar spaces in which every one point set is $G_\delta$ but space is not first countable

In an exercise from Munkres-Topology Article 30 the author writes that there is a very familiar space which is NOT first countable but every point is a $G_\delta $ set. What is it? Though there are ...
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1answer
37 views

If $(X,d_1)$ and $(X,d_2)$ two connected metric spaces if only if $X\times Y$ is connected metric space

$(X,d_1)$ and $(X,d_2)$ are two connected metric spaces if and only if $X\times Y$ is a connected metric space with metric $$ D((x_1,y_2), (x_2,y_2)) = \max(d_1(x_1,x_2),d_2(y_1,y_2)).$$ I know that ...
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2answers
34 views

Interior of a set in a metric space

if $E$ is a metric space nd $B\neq E$ how to prove that: $$\overset{\circ}{B}=\bigcup_{n=1}^{\infty} (\{x\in E, d(x, E\setminus B)\geq \frac1n\})$$ i don't know how to start
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3answers
51 views

Given a topological space $X$, why are both $X$ and $\emptyset$ open and closed? [duplicate]

I think this is a basic question, but it's hard to wrap my head around.
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1answer
51 views

Proof that $\{\,\left]a,\infty\right [\mid a\in\mathbb{R}\,\}\cup\{\mathbb{R} \}\cup\{\emptyset \}$ is topology of $\mathbb{R}$

Proof that $\mathcal{T}:=\{\,\left]a,\infty\right [\mid a\in\mathbb{R}\,\}\cup\{\mathbb{R} \}\cup\{\emptyset \}$ is topology of $\mathbb{R}$. I have slight trouble on writing this down.. I'll first ...