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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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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12 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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37 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
24 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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31 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 ...
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1answer
20 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
47 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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20 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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23 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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46 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
47 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 ...
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2answers
21 views

The product topology :It's definition and coarseness

Let A be a finite set of topological spaces $\ X_\alpha $ set. Now let us consider the product set of this topological spaces P=$\ \prod_\alpha X_\alpha $. Now a topology in P , with all sets of ...
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37 views

Can binary ring for homology make life easier?

Do you know of a proof which uses homology to demonstrate a property about a topological space which is made easier (or even possible) because they work over $\mathbb{Z}/2\mathbb{Z}$ instead of ...
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31 views

Two questions on Munkres -Topology

I have two questions: If $X$ is a countable product of spaces having countable dense subsets then does $X$ have a countable dense subset? Let $X$ $=\prod_{i=1}^\infty X_i$ .Let $D_i$ denote the ...
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1answer
15 views

Example where Alexandroff compactification $X^*$ is connected but the initial space $X$ is not

Let $(X,\tau)$ be a topological space that admits a one-point compatification $(X^{*},\tau)$ (Alexandroff compatification). I know that if the space $X$ is connected, then $X^*$ is connected as well. ...
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202 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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1answer
17 views

For $U\subseteq Y\subseteq X$, prove that $U$ is open in $Y$ iff there is a $V\subseteq X$ such that $U=Y\cap V$

Let $(X,d)$ be a metric space, with $Y$ a subset of $X$. How do I prove that a subset $U\subseteq Y$ is open in the metric space $(Y,d|_{Y\times Y})$ iff there exists an open subset $V$ of $X$ ...
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1answer
40 views

Is there any result on the “counting” of minimal atlas?

Take a differentiable manifold $M$. Define $\eta(M)$ as $\min\{\#\mathfrak{A} \mid \mathfrak{A} \text{ is an atlas for $M$}\}$. For example, if $M=S^n$, we have that $\eta(M)=2$, since $S^n$ is ...
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24 views

Prove that all three metrics induces the same topology on $X_1\times X_2$

Prove that if $(X_1,d_1)$ and $(X_2,d_2)$ are metric spaces on $X_1\times X_2$ and metric $d:(X_1\times X_2)\times (X_1\times X_2)\rightarrow R$ is defined in following way: ...
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1answer
18 views

what is non trivial basis for cofinite topology on non empty set $X$ [on hold]

what is non trivial basis for cofinite topology on non empty set $X$??? when $X $ is infinite set.
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25 views

To prove given set is basis for topology on $\mathbb{Z}$ [duplicate]

An arithmetic progression in $\mathbb{Z}$ is a set $A_a,_b=\bigg\{\dots,a-2b,a-b,a,a+b,\dots\bigg\}$ with $a,b\in\mathbb{Z}$ and $b\neq0.$ prove that the collection of arithmetic progressions ...
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33 views

Munkres Topology Article -30 Problem 5

Show that a metrizable space with a countable dense set has a countable basis. My try: Let $X$ be a metrizable space with a countable dense set $D$. Consider for each $n\in \Bbb ...
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138 views

Show that every compact metrizable space has a countable basis

Show that every compact metrizable space has a countable basis. My try: Let $X$ be a compact space and metrizable. Now for each $n\in \Bbb N$; I can consider the open cover $\{B(x,\frac{1}{n}):x\in ...
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70 views

Is there a nice open set proof that multiplication is continuous?

For students in a first course in analysis or topology, proving that certain function are continuous can be very tricky. However, some proofs which are difficult for students to prove using the ...
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1answer
20 views

different definitions of a subnet

The classical definition of subnet seems to be that $\Psi: J\to X$ is a subnet of $\Phi: I\to X$ if there exists a monotone, final map $h: J\to I$ s.t. $\Psi = \Phi\circ h$. I found another definition ...
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36 views

Map from circle to real line

I am asked to show that, for any continuous $\phi:\;S^1\to\mathbb{R}$ where $S^1=\{ \|\mathbf{x}\|=1,\;\mathbf{x}\in\mathbb{R}^2\}$, there exists $\mathbf{z}\neq 0$ such that: ...
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Cluster points and the sequence 1,1,2,1,2,3,1,2,3,4,1,…

I am working on a problem in analysis. We are given a sequence $x_n$ of real numbers. Then a definition: A point $c \in \mathbb{R}\cup{\{\infty, -\infty}\}$ is a cluster point of $x_n$ if there is a ...
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How to define “being inside of something” in the context of topology?

I'm a Psychologist and Neuroscientist with interest in math and I just started reading about Topology. I have to say it's not easy to grasp the concepts without a practical example, so I'm trying to ...
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1answer
37 views

$T^2-D$ does not retract to the boundary $\partial D$

First of all: yes, there is already a post about it, but I missread retract as strong deformation retract and wanted to know if this solution is right if we really do assume the stronger assumption of ...
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1answer
28 views

Product topology and projection mappings.

Let us consider two topological spaces $X$ and $Y$. Now let us consider projection mappings $p_1$ and $p_2$ defined from the product set of $X$ and $Y$ to $X$ and $Y$ respectively .The lecture notes I ...
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(i) $\{(x,y) \in \mathbb{R}^2 |\;xy = 1\}\,\bigcup\, \{(x,y) \in \mathbb{R}^2 |\;y = 0\}$ is not connected

I need to understand the following (i) $\{(x,y) \in R^2 |\;xy = 1\}\;U \{(x,y) \in R^2 |\;y = 0\}$ is not connected however (ii) $ Y = \{(x,y) \in R^2 |\;x^2 + y^2 < 1\}\;U \{(x,y) \in R^2 |\;y ...
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1answer
40 views

Homogeneous space minus a point

If $X$ is homogeneous and $p\in X$, then is $X\setminus \{p\}$ necessarily homogeneous? This seems to work with all the simple examples I've tried. I would be interested in any counterexamples. Or ...
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63 views

When is a diffeomorphism analytic?

I've read somewhere "a class $C^\infty$ diffeomorphism is said to be analytic" but I forgot to write down where I read this and now I can not find it, which makes me wonder if it's true? I'm working ...
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2answers
36 views

Any necessary and sufficient condition(s) for closure of an open ball to be the corresponding closed ball?

Let $(X,d)$ be a metric space, $a \in X$, and $\delta$ be a positive real number. Then the open ball $B(a;\delta)$ is defined as $$B(a;\delta) \colon= \left\{ \ x \in X \ \colon \ d(x,a) < \delta ...
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1answer
20 views

Regarding part of proof of proposition: Any topological group $(G, \tau)$ which is a $T_1$-space is also a Hausdorff space.

Proposition: Any topological group $(G, \tau)$ which is a $T_1$-space is also a Hausdorff space. Part of Proof: Let $x$ and $y$ be distinct points of $G$. Then $x^{-1}y \neq e$ (identity ...
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Is the set of all topological spaces bigger than the set of all metric space?

I was wondering right that since the notion of a topology is much more general than that of a metric, and that "neighborhodness", if you will, and the concept of continuity, is generalized by the ...
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2answers
26 views

Finding closures on $\mathbb R$ over a specific topology

I have the following topology over $\mathbb R$ $$ T = \{\emptyset\} \cup \{G\subseteq \mathbb R: \mathbb Q \setminus G \text{ is finite}\} $$ How could I study the closure of $\mathbb Q$ and $\mathbb ...
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1answer
29 views

Bijection bewteen $(-1,1)$ and $\{(x,y)\in\mathbb{R}^2:y=x^3\}$

I am trying to show that $(-1,1)$ and $\{(x,y)\in\mathbb{R}^2:y=x^3\}$ are homeomorphic, with the standard metrics. I cant see how to define a bijection.
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Relations between cluster points of nets and types of accumulation points of sets

Let $X$ be a topological space, $(x_\alpha)$ a net in $X$ and $A \subseteq X$ an arbitrary subset. The point $x \in X$ is a cluster point of $x_\alpha$ if for every neighborhood $U$ of $x$ the net ...
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Definition of topological space

The definition of a topological space is a set with a collection of subsets (the topology) satisfying various conditions. A metric topology is given as the set of open subsets with respect to the ...
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Sphere homeomorphic to interval times space

Let $Y$ be any topological space. In my notes I found the exercise to show that: $I \times Y \approx S^n $ via a homeomorphism is not possible, where $S^n$ denotes the $n$-sphere and $I$ the unit ...
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37 views

Postnikov tower of a product

Let $X$ and $Y$ be simply connected, locally finite CW-complexes and let $(X_i)_i$ and $(Y_i)_i$ be their Postnikov towers respectively. Is the Postnikov tower of $X\times Y$ given by the products ...
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1answer
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Sequentially compact iff every countably infinite subset has an infinite subset that has an $\omega$-accumulation point?

Let $X$ be a topological space and $A \subseteq X$ a subset. $A$ is called sequentially compact iff every sequence in $A$ has a convergent subsequence with limit in $A$. A point $x \in X$ is an ...
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1answer
32 views

Are all the subsets of $\mathbb{Z}$ closed or open (or neither) in $\mathbb{Z}$?

At each integer $n$, $B_r(n)=\{n\}$ for small $r$, so $B_r(n)=\{n\} \subset \mathbb{Z}$. Since any subset is a union of some integers, does this imply that all subsets are open? Also, since there is ...
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Why do we say a level 1 Menger Sponge has 5 holes? [on hold]

I've heard that a level 1 Menger Sponge has 5 holes, but what is the justification for this? I can understand starting with a hole down the center, and making 4 more to meet it from the sides, but ...
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1answer
50 views

Proving a topology is not induced by a metric

I'm reading a proof where it requires to show that a topology is not induced by a metric. My question is: What does it mean for a topology to be induced / not induced by a metric?
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23 views

Is preimage of closure equal to closure of preimage under continuous topological maps? [duplicate]

Let $f:X \rightarrow Y$ be a continuous map of topological spaces and $B \subseteq Y$ Is it true that $f^{-1}(\overline{B})=\overline{f^{-1}(B)}$?
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31 views

Construction of a continuous function

Given two sets $x = \{ a_1, a_2, a_3, a_4 \}$ and $y = \{ \emptyset, x, \{ a_1, a_2, a_3\}, \{ a_3 \}, \{ a_3, a_4 \} \}$, where $y$ is a topology defined on $x$. How could we construct a continuous ...
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1answer
16 views

Continuity on the parameters of the intermediate value theorem

Let $X$ be a compact metric space (feel free to impose more conditions as long as they're also satisfied by spheres) and $F : X \times [0, 1] \to \mathbb{R}$ a continuous function such that $F(x, 0) ...
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1answer
37 views

Prob. 10 (d), Sec. 19 in Munkres' TOPOLOGY, 2nd ed: How to show that this map is open?

Here's Prob. 10, Sec. 19 in the book Topology by James R. Munkres, 2nd edition: Let $A$ be a set; let $\{X_\alpha \}_{\alpha \in J}$ be an indexed family of spaces; and let $\{ f_\alpha ...