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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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
37 views

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

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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1answer
34 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
29 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
18 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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1answer
225 views
+50

Hatcher 2.1.10…

Hatcher asked a question in chapter $2$ (a) Show the quotient space of a finite collection of disjoint $2$-simplices obtained by identifying pairs of edges is always a surface,locally homeomorphic ...
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0answers
44 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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27 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
24 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
40 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
17 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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1answer
17 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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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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1answer
62 views

Prove that the following statements are equivalent characterizations of continuity

Let $f: (X,d) \rightarrow (Y, d')$ be a function. Prove that the following are equivalent: $f$ is continuous . For every $A \subset X$, $f(cl(A)) \subset cl(f(A))$. For every closed set $B$ in ...
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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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2answers
28 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
544 views

Cantor set in base 3.

I'm trying to prove the cantor set $C$ is equivalent to the set of all numbers with ternary expansion of $2$'s and $0$'s. That is: Let $A_0=[0,1]$. $A_n$ is defined to equal $A_{n-1}$ with it's ...
4
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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 ...
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2answers
59 views

If two nested open sets have the same nonempty boundary, are they the same set?

Let $(X,d)$ be a metric space. Let $B_\epsilon(x)$ be the open ball of radius $\epsilon$ centered at $x$. For $x\in X$ and $\epsilon>0$, suppose that $V$ is an open set in $X$ with $V\subseteq ...
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1answer
69 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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0answers
39 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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16answers
34k views

Best book for topology?

I am a graduate student of math right now but I was not able to get a topology subject in my undergrad... I just would like to know if you guys know the best one..
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2answers
25 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 ...
1
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1answer
179 views

In a standard metric space…what does | | mean (is it the absolute value or something more)?

We have a standard metric space defined as: ($\mathbb{R}$,d)= ($\mathbb{R}$, | |) $d(x,y)=|x-y|$ Does | | in first sentence always mean that we must do $|x-y|$; so that we look only at the distance ...
3
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0answers
77 views

Quotient space of $\Bbb R^2 / {\sim} \approx T^2$ defined by $\mathbf x \sim \mathbf y \iff \mathbf x=\mathbf y A^n$

Below is my homework problem. But I cannot even think of a vague idea for a proof. Can anyone give some idea for the proof? Let $A$ be an invertible $2 \times 2$ matrix with entries in $\mathbb ...
3
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2answers
55 views

If $d(x,A)=0\forall x\in X$ for some subset $A$ of $X$, does it follow that $A$ is dense?

If $d(x,A)=0 \:\:\forall x\in X$ for some subset $A$ of $X$ then $A$ is dense in $X$, right? Once I did one problem which says $d(x,A)=0\Leftrightarrow x\in \bar{A}$ so by the condition here we get ...
1
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1answer
76 views

A question about generic points.

Let $X$ be a topological space and let $x$ be a point in $X$. In Hartshorne's book, I saw the the following definition of a generic point: $x$ is a generic point of $X$ if $\{x\}$ is dense in $X$. ...
0
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1answer
91 views

If $p(z,w)=a_0(z)+a_1(z)w+\dots +a_k(z)w^k$ are non constant polynomial.

$p(z,w)=a_0(z)+a_1(z)w+\dots +a_k(z)w^k$ where $a_i(z)$ are non constant polynomials in complex variables with $k\ge 1$. I need know if $$\{(z,w):p(z,w)=0\}$$ which of these are true or false: ...
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2answers
143 views

An open cover that is not locally finite

I could not understand why example $13.4$ is not locally finite. Can you give me an explanation please.
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0answers
32 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 ...
3
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1answer
107 views

What is an overlap?

I want to ask what an overlap is. My teacher said that for example $1$: Everything is an overlap hence it is not locally finite. For example $2$, it doesnt overlap. Please teach me these two ...
0
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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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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
29 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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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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4answers
480 views

How to show that continuous functions between metric spaces agree on a closed set

Let $(X,d)$ and $(Y,d')$ be metric spaces, and let $D$ be a dense subset of $X$. Show that: If $f:X\to Y$ and $g:X\to Y$ be continuous, then the set $\{x\in X\mid f(x)=g(x)\}$ is closed.
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 $\forall q|d(p,q)<r$, ...
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1answer
47 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 ...
4
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3answers
275 views

Topological boundary vs geometric boundary

Let $M_1=B((0,0),1)=\{(x,y) \mid x^2+y^2<1\}$ $M_2=\{(x,y) \mid x^2+y^2\le1\}$ What are the interior of $M_1$ and $M_2$ ? And what are the boundary of $M_1$ and $M_2$ ? How do I find them? ...
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1answer
36 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 ...
2
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1answer
82 views

Problems on topological and metric spaces.

I need help on the following problems: Q1: Consider the map $f:(\Bbb R\times\Bbb R,\tau)\to(\Bbb R,\gamma)$ given by $f(x\times y)=|x-y|$,where $\tau$ is the product topology and $\gamma$ is the ...
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2answers
26 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, ...
3
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2answers
55 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
34 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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2answers
218 views

Tweaking the axioms of a Topological Space, what are the consequences?

A topological space is a set $X$ together with a topology $\tau$ (a collection of open subsets) such that. $\emptyset\in \tau$ and $X\in \tau$. The intersection of a finite number of sets in ...
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2answers
260 views

Looking for a (nonlinear) map from $n$-dimensional cube to an $n$-dimensional simplex

I am looking for a (nonlinear) map from $n$-dimensional cube to an $n$-dimensional simplex; to make it simple, assume the following figure which is showing a sample transformation for the case when ...
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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 ...
4
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0answers
74 views

Properties of King's Dream fractal

My question is focused on the King's Dream fractal, which can be defined as follows (nice pictures can be found here) : $$ \Omega = \{f^n(0.1,0.1) \;\vert\; n \in \mathbb N \} \quad ...
0
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1answer
19 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 ...
2
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1answer
24 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 ...