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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Showing a set is closed with sequences

$G \subset R^n$ Let $\{x_k\}_{k\in N}$ a convergent sequence in $G$, ($x_k \in G$ for every $k$). $lim_{k \to \infty} x_k = a$ lays in $G$. Show that G is a closed set. Help please, I have already ...
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3answers
971 views

Prove that $S = \{(x,y)\in \Bbb R^2 : x>0, y>0\}$ is open.

Prove that $S = \{(x,y)\in \Bbb R^2 : x>0, y>0\}$ is open. Can anyone help me, I need to prove this statement only by the use of a ball.
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1answer
58 views

Let $ f:(X, d) \mapsto (Y,d) $ be an mapping such that $ Graph (f) $ is connected. [duplicate]

Where $ X $ is connected. Does it imply $ f $ to be continuous?
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1answer
33 views

Figuring out parameters

I need to figure out a parameter to satisfy the following conditions: $H(\frac{1}{2}, 0) = 0$ $H(1,0) = 1$ $H(0,1) = 0$ $H(1,1) = 1$ for $H(s,t)$. I have been at it for hours and can not figure ...
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1answer
44 views

Question about the fundamental group of simplicial complex.

Suppose we have a simplicial complex G that is finite connected. (1)The fundamental group of G is finite; (2)The universal cover of G is compact. Question: Can (1) implies (2)? Thanks.
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0answers
67 views

How to deduce the results of response time by this trajectory approach?

First, we denote this: And And we get this right property( $last_i$ means the last node on $τ_i$): And: $Smin_i^h$ = $\sum_{h'=first_i}^{h-1} ({C_i^{h'} + L_{max})}$ $Smax_i^h$ = ...
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0answers
68 views

Very Simple Universal Covering problem

A space $X$ is constructed from two disjoint copies of $RP^3$ and a copy of the unit interval $I$ by gluing one end of $I$ to a point of one copy of $RP^3$, and gluing the other end of $I$ to the ...
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1answer
52 views

Topology Qual Question 1

I was working on some old topology qualifying exam problems and got stuck on this one. Any help would be greatly appreciated. Consider the smallest equivalence relation on $X = S^1 \times S^1$ such ...
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2answers
198 views

Past Topology Qualifying Exam Question

When practicing old qualifying exam problems, I had trouble with this one. Thanks for any help! Is it true that if the $1$-point compactifications of two locally compact Hausdorf spaces $X$, $Y$ are ...
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1answer
119 views

How to show a topological space is compact and if it's from a metric or not?

We have $X$ is an infinite set and $\tau=\{U\subset X:X \backslash U$ is finite or is all of $X \}$. Till now, I have proved that $\tau$ is a topology and is not Hausdorf, but how can I show if it's ...
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1answer
65 views

Topology Question on Connectedness

I have no idea how to do this problem. Any suggestion would be great Let $f : X \rightarrow Y$ be a continuous map. Suppose that $Y$ is connected, and that $f^{-1}(y)$ is also connected, for each ...
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115 views

general topology exercise 777 [closed]

Let f : $S^1$ → R be continuous, where $S^1$ is the unit circle in $R^2$. (a) Show that there is a point z ∈ $S^1$ such that f(z) = f(−z). [z = (x; y), −z = (−x;−y)]\ (b) Show that f is not ...
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1answer
201 views

Quotient function [closed]

Let $f : X \to Y$ be a quotient map, and $g : X \to Z$ a continuous function such that $g(x_1) = g(x_2)$ whenever $f(x_1) = f(x_2)$. Show that there exists a unique continuous function $h : Y \to Z$ ...
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102 views

Examples of $\kappa$-Fréchet-Urysohn spaces.

We say that A space $X$ is $\kappa$-Fréchet-Urysohn at a point $x\in X$, if whenever $x\in\overline{U}$, where $U$ is a regular open subset of $X$, some sequence of points of $U$ converges to ...
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1answer
64 views

Compactification of a discrete space using ultrafilters.

I want to show for the collection of ultra filters on a (non-empty) set $A$, $U(A)$. That $U(A)$ is compact where the topology is derived from the base $U_B = \{F\in U(A)|B\in F\}$. Seeing as $A$ can ...
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0answers
339 views

Tychonoff theorem (2/2)

In following I would like to post a proof of Tychonoff theorem using filters of closed sets. I would be grateful if you could find any mistakes in my proof (also if you read and don't find any ...
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1answer
50 views

Show that the set is convex

A set $Z$ is convex when $z_1,z_2 \in Z\Rightarrow (1-h)z_1+hz_2\in Z$ for all $h\in [0,1]$. Show that if $X$ and $Y$ are convex than $X*Y=\{(1-t)x+ty\;|\;x\in X,y\in Y, t\in [0,1]\}$ is convex.
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1answer
173 views

is infinite connected components still true

my assignment has this question, given a topological space X with finite many connected components, a function $f:X\rightarrow Y$ is continuous if and only if it is continuous on each components. Is ...
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0answers
489 views

The Cantor Space as $\{0,1\}^{\mathbb{N}}$ and as $[0,1]$.

The Cantor-Space is defined as the space of all infinite binary sequences, i.e. the space $\{0,1\}^{\mathbb{N}}$. It has a natural metric, $$ d(x,y) = \inf\{ 2^{-|w|} : w \in pref(x) \cap pref(y) \} ...
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1answer
99 views

Compact sets of the complex plane with countable boundary

Suppose $E$ is a compact set of the complex plane and the boundary of $E$ is a countable set. How does one prove that $E$ is equal to its boundary?
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1answer
68 views

$f^{-1}(U)$ is regular open set in $X$ for regular open set $U$ in $Y$, whenever $f$ is continuous.

Let $f$ be a continuous function from space $X$ to space $Y$. If $U$ is regular open set in $Y$, it it true that $f^{-1}(U)$ is a regular open set in $X$?
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1answer
148 views

Is this set dense in $H^1(\Omega)?$

Is $$V_1 = \{v \in H^1(\Omega) \;:\;f(v) = 0 \text{ on } \partial \Omega\}$$ dense in $H^1(\Omega)$ with the same norm as $H^1(\Omega)?$ Here $f$ is some linear functional so that $V_1$ is also ...
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1answer
633 views

Does there exists a continuous surjection from $\mathbb{R}$ to $\mathbb{R}^2$?

I constructed a bijection by using decimal expansions of two real numbers and taking numbers 1 by 1 consecutively. (It took me hours to come up with this). I remember someone saying appealing to some ...
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1answer
128 views

Are WOT/SOT topologies hereditarily separable?

Just out of curiosity, Are weak and strong operator topologies on $B(H)$ hereditarily separable? In other words, if $S$ is a subset of $B(H)$, where $H$ is a separable Hilbert space, is $S$ ...
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113 views

Help in understanding product topology

Let $X_i$ be topological space. Let $O_i \subseteq X_i$ denote open set and let $C_i \subseteq X_i$ denote closed set. Let $O$ be open set in $X = \prod_i X_i$ and let $\pi_i : X \to X_i$ be ...
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1answer
55 views

Does this result apply for locally injective function?

Definition: Locally Injective Function Let $F : ℝ^{r+1}→ℝ^{r+1}$ be a continuous function, and let $a∈ℝ^{r+1}$. We say that $F$ is locally injective (or locally one-to-one) at $a$ if there exists a ...
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3answers
108 views

Topology induced by metric

Get an example of a metric on a countable set that not generates the discrete topology. I think it may be a set in this way $0 \cup\{1/n:n\in\mathbb N\}$ with the metric $d(x,y)= \vert x-y \vert$ ...
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2k views

When is the closure of an intersection equal to the intersection of closures?

Obviously $\overline{\bigcap A_{\alpha}}\subseteq\bigcap\overline{A}_{\alpha} $, but when is the reverse inclusion true? Can you give some properties of the underlying space that would guarantee ...
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1answer
194 views

Quotient map question

I'm having trouble demonstrating this map is a quotient map, $f: \Bbb R^2 \to \Bbb R^2$ defined by $(x,y)\mapsto (x \cos(y),x\sin(y))$ with $x \neq 0$. Showing the map is surjective isn't difficult ...
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1answer
330 views

Klein bottle homeomorphic to union of Möbius strip

I'm having trouble showing that the Klein bottle defined as a quotient space of $I^2$ with relation $(x,-1)R(x,1)$ and $(-1,y)R(1,-y)$ is Hausdorff and that it can be expressed as $X\cup Y$ where ...
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2answers
77 views

Manifold path components are open

Can someone explain why the path components of a manifold are open? I'm a little confused at how to demonstrate this fact and it would obviously help me understand manifolds better.
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315 views

Does Seperable + First Countable + Sigma-Locally Finite Basis Imply Second Countable?

A topological space is separable if it has a countable dense subset. A space is first countable if it has a countable basis at each point. It is second countable if there is a countable basis for ...
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4answers
90 views

Measure nonzero implies dense on a rectangle

This would be a very handy lemma for me but I have been unable to prove it thus far. If $S \in \mathbb{R}^n$ is bounded and is not of measure zero, then there exists a rectangle $R$ such that $S$ ...
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140 views

Topological proof to show a complement set of a set is polygonally connected

I started learning topology recently, so I'm very new to this. The question is as follows: Given the set $A = \{ (x,y)\;;\;x, y\in\mathbb{Q} \}=\mathbb{Q}^2$, show that the complement set ...
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1answer
74 views

For each continuous $g:X\to [0,1]$, $g(a_n)\to g(a)$, can we deduce $a_n\to a$?

Assume $(X,\mathcal T)$ is a Tychonoff space. $(a_n)$ is a sequence in $X$. $a\in X$. for each continuous function $g:X\to [0,1]$, $$g(a_n)\to g(a)$$ Is there an elementary proof for $$a_n\to a$$ ...
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1answer
64 views

Metacompactness of the Euclidean space

Does anyone know how to prove that every Euclidean space is countably metacompact? In particular, my interest is in $R^2$. Thanks,
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1answer
93 views

About interior, closure and etc in $\mathbb{R}^2$

Given this set $A$: $$A =\left\{ \, (x,y) \mid x = 1/n, \ |y| \le n, \ n \in \mathbb{N} \, \right\} \subset \mathbb{R}^2; $$ I'd love to find the interior, closure, set of limit points, set of ...
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3answers
246 views

Separable space and countable

How do I show that every collection of disjoint open sets in a separable space must be countable? I am studying separable from this book and this was stated, and the prove was left to the reader. I ...
3
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1answer
141 views

Dense subspaces

How does one go about proving the following statements? (a) $\operatorname{Lip}[a,b]$ functions are dense in absolutely continuous functions on $[a,b]$ under the variation norm - (Another doubt: what ...
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33 views

Prove that equivalent distances define the same topology on a given set. [duplicate]

How do I answer this: I'm not looking at open balls so I'm not to sure where to start. Please help: Prove that equivalent distances define the same topology on a given set.
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1answer
82 views

How to show that a continuous map on a compact metric space must fix some non-empty set.

Suppose $(X,d)$ is a compact metric space and $f:X\to X$ a continuous map. Show that $f (A)=A$ for some nonempty $A\subseteq X.$ I start this by supposing that $A_0:=X$ and $A_{n+1}:=f(A_n)$ for ...
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489 views

Inductive Limit Topology and First Countability

Motivation: I am doing functional analysis on locally convex spaces for the first time and I would like to know when I am allowed to characterise limit points and continuity sequentially. (This may ...
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129 views

are “all nets in $X$” well defined?

Denote $f:X\to Y$ as a function between topological spaces $X$ and $Y$. One good way for determining whether $f$ is continuous is to check the following statement. $f$ is continuous iff for every ...
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1answer
329 views

Intermediate value property and closedness of rational level sets implies continuity

Suppose $f$ satisfies the intermediate value property, i.e. if $f(a)<c<f(b)$, then there exists $a<x<b$ such that $f(x)=c$ and for every rational $r$, $S_r$ such that $f(x)=r$ is a closed ...
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1answer
37 views

The topology produced by indexing subsets of a topological space.

Let $(X,\mathcal T)$ be a topological space. For each $A\subseteq X$, let $i_A:I_A\to A$ be an onto function. Let's define: $$f:\mathcal P(X)\to \mathcal P(X)$$ where $f(A)$ contains a point $x\in ...
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2answers
95 views

How to prove that the surface of genus 2 can be represented as an octagon

I've been trying to show that the surface of genus 2 can be represented by appropriately identifying the edges of a regular octagon. I think have managed to work out the way to identify the edges but ...
5
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1answer
78 views

Is $(A \times B)^\epsilon \subseteq A^\epsilon \times B^\epsilon$?

While working on a problem related to my research, I had the following query. It pertains to product spaces: The Question: Let $(X,d_X)$ and $(Y,d_Y)$ be two Polish (Complete separable metric) ...
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3answers
97 views

Is a topological space with a minimal dense subset, finite?

$(X,\mathcal T)$ is a topological space and $A$ is dense in it and for each dense $B$, we have: $$B\subseteq A\to B=A$$ Is $X$ finite?
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150 views

Showing $\mathbb{Q}$ is homeomorphic to $\mathbb{Q}^2$

Let $\mathbb{Q}$ be set of rationals, as ralative topology to $\mathbb{R}$. How to show $\mathbb{Q}$ is homeomorphic to $\mathbb{Q}^2$? More general, if $T$ is countable dense space(no isolated ...
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1answer
105 views

Convergent sequence in product space implies mappings converge

I asked this question yesterday. I thought I had the forward direction figured out but now I've lost confidence. Let $x_1,x_2,\ldots$ be a sequence of points of the product space $\prod ...