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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2answers
18 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
42 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.
0
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1answer
40 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 ...
0
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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 ...
1
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1answer
35 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 ...
0
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1answer
29 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 ...
1
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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. ...
7
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2answers
135 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 ...
0
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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 $X$ ...
6
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1answer
37 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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2answers
23 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: ...
0
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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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2answers
24 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 ...
1
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1answer
32 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 ...
6
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4answers
136 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 ...
0
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2answers
69 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 ...
1
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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 ...
3
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2answers
35 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: ...
1
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2answers
28 views

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 ...
12
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4answers
701 views

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 ...
1
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1answer
36 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 ...
0
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1answer
27 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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4answers
36 views

(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 ...
1
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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 ...
0
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0answers
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 ...
2
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2answers
35 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 ...
2
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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 ...
16
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5answers
738 views

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 ...
2
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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 ...
0
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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.
0
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0answers
21 views

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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2answers
37 views

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 ...
7
votes
1answer
47 views

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 ...
0
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1answer
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 ...
1
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1answer
15 views

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 ...
1
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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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0answers
26 views

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 ...
1
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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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0answers
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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1answer
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 ...
2
votes
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) ...
1
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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 ...
0
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1answer
15 views

Tychonoff space with unique compactification and 3 disjoint non-compact closed subsets

Prolog : The only compactification of a non-compact normal space $S$ is the one-point (Alexandroff) compactification IFF whenever $A,B$ are disjoint closed subsets of $S$, at least one of $A,B$ is ...
0
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0answers
25 views

What does the “closure of its graph” mean

I Am confused with various terminologies spelled out the same but meaning very differently depending on the situations. There are just too many. Here, I only understand the "closure" in the ...
1
vote
1answer
27 views

Is $\{(1,0),(0,0)\}\cup\bigcup_{n\neq1}\{(x,\frac{1}{n}):x\in\Bbb{R}\}$ locally connected?

Let $X=\{(1,0),(0,0)\}\cup\bigcup_{n\neq1}\{(x,\frac{1}{n}):x\in\Bbb{R}\}$. Determine whether or not $X$ is locally connected and find its components. Well, I know that a space $X$ is said ...
0
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0answers
18 views

Show that $h$ is homotopic to the identity map relative to $C$.

This is problem 5.3 and 5.4 in Armstrong's Basic toplogy. They are very much connected and i have solved problem 3. 3: Let $D$ be the disc bounded by $C$, i.e. $S^1$, parametrize $D$ using polar ...
0
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1answer
67 views

Cellular homology of the real projective space $\mathbb R P^n$

I've been able to calculate the cellular homology of $\mathbb R P^2$ but I'm struggling to do the same for higher dimensions. My problem is that I don't exactly see how one get to the result $d_i: ...
2
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0answers
35 views

How to prove this criteria of differentiability? [duplicate]

Let $f: I \to \mathbb{R}$ continuous and $a\in \operatorname{int}(I)$. Suppose that there is $L\in\mathbb{R}$ such that $$\lim \frac{f(y_n)-f(x_n)}{ y_n-x_n}=L$$ for all sequences $(x_n)$ and $(y_n)$ ...
1
vote
1answer
101 views

Subsets of the reals when the Continuum Hypothesis is assumed false

If one assumes that the continuum hypothesis is false then there are subsets of the reals of intermediate cardinality, uncountable but smaller than the continuum. What can be said about the necessary ...
1
vote
0answers
35 views

Proving open neighbourhood in topology

Let $X$ be the set $(\mathbb{R}\backslash \mathbb{N}) \cup \{1\}$. Define a function $f:\mathbb{R} \rightarrow X$ by $$ f(x) = \left\{ \begin{array}{ll} x & \mbox{if $x \in ...