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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2
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1answer
25 views

Show that $\lbrace 1-\frac{1}{n} \rbrace_n$ does not converge in the Sorgenfry topology.

Consider $\{1-\frac{1}{n}\}_n=\{0,\frac{1}{2},\frac{2}{3},\frac{3}{4},\cdots\}$. If $\{1-\frac{1}{n}\}_n$ converges, then $\{1-\frac{1}{n}\}_n \rightarrow 1$. If it converges, then, by definition, ...
1
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0answers
20 views

Continuous Extension Mapping

Let assume $A$ is a dense subspace of a topological space $X$ and $f$ is a continuous mapping of $A$ to a regular space $Y$. the question is :- Show that the mapping $f$ has a continuous extension ...
0
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1answer
17 views

Question about Helly Space.

I have two question about Helly Space:- Are Helly space is compact? how to show that Helly space contains a subspace homeomorphic to the discrete space $D(\mathfrak{c})$ and subspace homeomorphic to ...
1
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2answers
34 views

An example of a not locally compact space in $\mathbb R^2$

Are the two subspaces $X$ and $\operatorname{cl}(X)$ of Euclidean space $\mathbb R^2$ locally compact? $$X = \{(x,\sin 1/x) \mid 0 < x \le 4\}\cup\{(x,\sin 1/x) \mid -4 \le x \lt 0\} \cup ...
0
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0answers
24 views

What does the positive minimal radius mean for an open cover?

The formal definition seems a bit confusing, any more words or pictures to explain it? Definition. Let $U$ be an open cover of $(M,d)$. We say that $U$ has positive minimal radius when there ...
2
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0answers
15 views

Limit topology of a sequence of topological vector spaces

Under which circumstances is the limit topology of an increasing sequence $E_0\subseteq E_1\subseteq E_2\subseteq\cdots$ of topological vector spaces, where the inclusion maps are linear and ...
1
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1answer
27 views

A problem in locally compact Hausdorff space

I am trying to solve the following problem. Let $X$ be locally compact Hausdorff and $Y$ be Hausdorff. (a) If $f: X \to Y$ is continuous and open map then show that $f(X)$ is locally compact. (b) ...
2
votes
1answer
29 views

are there different notions of 'boundary' in the manifold sense and the topological space sense?

this seemingly innocent question has been bugging me for quite a while. Lets give a minimal example: The unit circle S has no boundary considered as a manifold (all points have neighborhoods ...
2
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1answer
30 views

one point compactification of discrete space

Problem: What is the one point compactification $X^*$ of a discrete space $X$. In the case of $X$ being finite, $X$ itself is compact so the one point compactification would be merely $X$ $\bigcup$ ...
0
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2answers
41 views

Cardinality of a discrete set, in a separable space.

Given a separable space $X$, if $A$ is discrete subspace of $X$, then $|A|\leq 2^{\aleph_0}$. Some ideas?. It's similar to "jone's lemma", but without $A$ being closed. Whit what addiotional ...
1
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1answer
15 views

some statements on sum of two subsets of plane. open, closed etc .

$W=\{(x,y)\in\mathbb{R}^2: x>0,y>0\}$ $X=\{(x,y)\in\mathbb{R}^2: x\in\mathbb{R},y=0\}$ $Y=\{(x,y)\in\mathbb{R}^2: xy=1\}$ $Z=\{(x,y)\in\mathbb{R}^2: |x|\le 1,|y|\le 1\}$ $W+X$ is open ...
5
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3answers
124 views

Totally disconnected topologies on countable set.

Are there totally disconnected topologies $\tau$ on a countable set $X$ such that $(X,\tau)$ is not homeomorphic to one of the following? $\mathbb{N}$ with the discrete topology; one-point ...
2
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2answers
34 views

a question about connected set, how to know whether A is connected or not?

In the Euclidean plane $R^2$,consider the subset $$ A=\{(x,y)\in \Bbb R^2|\text{Either $x$ or $y$, but not both, is a rational number}\} $$ Is $A$ connected? Is $\Bbb R^2$\A connected? I have ...
0
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1answer
53 views

Is the definition of continuity in analysis a particular case of topological continuity?

Take a constant function and remove an open interval from it: $$f(x)= 1, \text{if $x\in(-\infty,0]\cup[1,\infty)$ }$$ This function shouldn't be continuous because at $0$ no right limit of the ...
4
votes
2answers
59 views

Show that the set of all complex numbers $z$ such that $|z| \leq 1$ is closed?

I'm working through Rudin's "Principles of Mathematical Analysis" on my own, so I don't want the full answer. I'm only looking for a hint on this problem. Rudin states without proof that the set $X = ...
3
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0answers
34 views

Topological proof for this set theory statement

Let $\mathcal{A}$ be an algebra of set (in a space $X$), such that any subcollection of disjoint sets in $\mathcal{A}$ is finite. Prove that $\mathcal{A}$ is finite. I already found a boring brute ...
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0answers
18 views

Exhaustion of a manifold by compacts

I searched for a proof of the following statement, but did not find one. I want to check if a proof I made is correct, or if I'm leaving out some detail and/or complicating things: Proposition: ...
2
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0answers
41 views

A question about compact sets: how to prove $g$ must be an isometry

Let $(X,p)$ be a compact metric space. Suppose that $g:X\rightarrow X$ is a function such that for all $x_1,x_2\in X$ we have $p(g(x_1),g(x_2))\geq p(x_1,x_2)$. Prove that, in fact, $g$ must be an ...
0
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2answers
50 views

Are $[0,1)\times [0,1)$ and $[0,1]\times [0,1)$ homeomorphic?

Are $[0,1)\times [0,1)$ and $[0,1]\times [0,1)$ homeomorphic? Not getting any idea how to start.
0
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1answer
41 views

What is the largest complete subspace of $(\mathbb{Q}, |\cdot|)$

For example $\left\{\frac{1}{n}\right\}\cup \{0\}$ is a complete subspace of $\mathbb{Q}$, but I am having trouble writing out the largest (in the sense of "$\subset$") complete subspace in ...
1
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1answer
20 views

a question about compact set, how to prove there exits f(y)=y [duplicate]

Let (X,p) be a compact metric space.Suppose that f X->X is a function such that, for all $x_1$,$x_2$ $\in$X, if $x_1\neq x_2$ then p(f($x_1$),f($x_2$))<$p($x1$,$x2$)$. Prove that there exits a ...
3
votes
2answers
40 views

One point compactification of $\Bbb{R}\setminus \{0\}$

What will be one point compactification of $\Bbb{R}\setminus \{0\}$? It looks like it will be union of two circles touching at a point. But do I write a Mathematical proof to justify my claim?
0
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2answers
20 views

$x$ x $1/x$ for $\epsilon$ $\gt 0$ has no $\epsilon$-neighborhood in $R_{+}$ x $R_{+}$

This is a problem from Munkres' Topology. Define the $\epsilon$-neighborhood of $A$ in a metric space $X$ to be the set $U(A, \epsilon) = ${$x$ | $d(x,A)$ $\lt$ $\epsilon$}. (d) Assume that $A$ is ...
1
vote
1answer
28 views

sequence of close and bounded sets in a prefect space

Suppose that$(E_n)$$_{n \in \mathbb N}$ be a sequence of closed and bounded sets in complete space $M$ such that $ E_{n+1} \subseteq E_n$ for all $ n \in\mathbb N$. If $\lim \operatorname{diam} E_n ...
1
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2answers
43 views

Frechet-Hausdorff theorem reference from J.L. Kelley used in proof that each probability measure is inner regular

Theorem: If $S$ is a complete, separable metric space, then each probability measure on it is inner regular. Proof: Since $S$ is separable, for each $n \in \mathbb{N}$ there exist countably many ...
0
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3answers
39 views

Let (X, d) be a metric space and A, B ⊂ X be two compact subsets. Show that A ∩ B is also compact

Question seems fine i just have a few doubts. Is it possible to just use the Heine Borel theorem? as both A and B are compact it implies they are both closed, so therefore their intersection is ...
1
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0answers
19 views

Is the translation of open and closed sets to some language non-antonym preserving?

Maybe more than one person though, before you were given the definition of closed set, that they were the sets that are not open, i.e. that the property of open and closed being antonyms were ...
0
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1answer
24 views

what is the definition of “two parallel copies of a surface S”

As indicated in the title, suppose $S$ is a surface with genus $g$, then what is the definition of "two parallel copies of S"?
1
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1answer
23 views

A problem in compactness in Euclidean Space using a special topology

Let $\mathscr U$ denote the usual topology on $R^2$ and consider the topology $\mathscr T = ${$U$ $\subset$ $R^2$ | $R^2 - U$ is a compact subset of ($R^2$, $\mathscr U$)} $\bigcup$ {$\emptyset$}} ...
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0answers
67 views

Is this a general structure for constructs?

Here a construct is a category where the objects are sets and the morphisms are structure preserving functions. Common examples are groups, graphs and topological spaces. As far as I can see there is ...
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0answers
13 views

Seperation of convex compact sets with affine halfspaces

Let $C_1,C_2,...,C_m$ compact convex sets s.t. $\bigcap C_i = \emptyset$. I want to show that in that case there exsist affine halfspaces $H_i$, such that for every $i=1,2...,m$, $C_i \subset H_i$ ...
1
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2answers
27 views

$\phi, \psi$ homeomorphisms on $U, V$ $\implies$ $\phi(U\cap V) \cong \psi(U\cap V)$?

Let $U,V \subset M$ be open subsets in some manifold $M$. Let $\phi, \psi$ be homeomorphisms on $U, V$ respectively. Is it true that we then have $\phi(U\cap V) \cong \psi(U\cap V)$?
-1
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2answers
20 views

Continuity of addition map with lower limit topology

Prove $f:\Bbb R\times\Bbb R \to\Bbb R$ (with lower limit topology on $\Bbb R$ in range and product topology on $\Bbb R\times\Bbb R$ from $\Bbb R$ with lower limit topology), where $f((x,y)) = x+y$, is ...
1
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2answers
47 views

Prove there exists dense open set

Let $G$ be an open set in $X$ and $D$ be a dense open set in $G$.Show there exists a dense open subset $V$ of $X$ such that $V\cap G=D$. Since $D$ is open in $G$, there exists $V$ open in $X$ ...
0
votes
1answer
48 views

Topological , Homeomorphic version of $|S \times S|=|S| $

Give example of a subset $A$ of $\mathbb R$ such that with respect to some topology , $ A$ is homeomorphic to $A\times A$ . In set theory ZF it is known to be equivalent to A.C. that for any ...
5
votes
1answer
126 views

algebra with topology homework problem

Hello Everyone, I have this homework problem, I'm going to share what i have so far, not sure if Im in the right path. First, I have: $$f \sim g \, \Leftrightarrow \,x_0 \in \mathbb{R^n}, \exists ...
0
votes
2answers
42 views

If two sets are separated, then any two subsets of those sets are also separated?

I want to prove that if two sets X and Y are separated, then subsets of those sets are also separated. The definition is that if X intersect Y closure is empty and X closure intersect Y is empty, the ...
1
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2answers
31 views

Closed set on topological space [duplicate]

This is a problem on topological spaces and continuous functions. If $f,g \to\mathbb{R}$ are continuous functions, then $T=\{x\in X: f(x)=g(x)\}$ is closed on X
0
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0answers
26 views

Question about dimension in Notherian spaces

Let $X$ be a Notherian topological space of finte dimension which is Kolmogorov (meaning that for two points $X$ there exists an open subset of $X$ containing one of them but not the other). This ...
3
votes
2answers
76 views

Is every metric space subspace of some connected metric space?

If the space itself is connected then we're done, but if not then I think we can extend our metric space to make it connected .I'm not sure whether this will work or not, but intuitively I think the ...
0
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2answers
33 views

Open connected subset of $ \mathbb R^2 $is path connected [duplicate]

Is open connected subset of $ \mathbb{R^2} $ is path connected?
0
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1answer
35 views

A set of real numbers whose limit points from a countable set

Construct a set of real numbers whose limit points from a countable set. Is the set you constructed closed? Is it compact? My example is $$G=\{1/n+1/m: n, m \in \mathbb N\}\cup \{0\}$$ and as ...
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0answers
16 views

When is a metrizable topological vector space locally bounded?

Consider a topological vector space $E$ with topology $\sigma$. Suppose that $E$ is metrizable, in other words, that there exists a metric $d$ on $E$ that induces the topology $\sigma$. One can then ...
0
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2answers
27 views

Klein bottle contains Mobius band

I read the following: "The Klein bottle contains a copy of the Mobius band". I assume this means that there is a subspace of the Klein bottle that is homeomorphic to the Mobius band. How do we obtain ...
0
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0answers
10 views

Orientability of Bordered Presentation and its Closure

How can the following claim be true? If $\Pi $ is a bordered presentation, then $\Pi ^c$ is orientable if and only if $\Pi $ is orientable. We know that $\Pi$ must contain border arcs. By ...
3
votes
2answers
45 views

Connected sum of projective plane $\cong$ Klein bottle

How can I see that the connected sum $\mathbb{P}^2 \# \mathbb{P}^2$ of the projective plane is homeomorphic to the Klein bottle? I'm not necessarily looking for an explicit homeomorphism, just an ...
1
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1answer
29 views

Show compactness of a set given by inequalities

Show that the subset $A=\{(x_1,...,x_n)\in\Bbb R^n |−1≤x_1 ≤x_2 ≤···≤x_n ≤1\}$ is compact. A is contain in an open cover as it is contained in $\Bbb R^n$. Therefore there exists a finite sub cover ...
2
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1answer
25 views

$t$-adic topology (on $\mathbb F_p(1/t)$)

Recently I found this interesting discussion about algebraically closed fields of positive characteristic. In the answer marked as the top answer, I read about the $t$-adic topology. The $t$-adic ...
0
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0answers
15 views

Performing an Excision on a Topological Surface

Recently I began a book on topology, but the concept of excision on a topological surface isn't clear; perhaps you, collectively, could help elucidate it. Suppose we have an arbitrary topological ...
5
votes
1answer
47 views

Theorem 3.54 (about certain rearrangements of a conditionally convergent series) in Baby Rudin: A couple of questions about the proof

Here's the statement of Theorem 3.54 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $\sum a_n$ be a series of real numbers which converges, but not absolutely. Suppose ...