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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Closed subspace of a compact topological space is compact

Let $X$ be a compact topological space, and $A$ a closed subspace. Show that $A$ is compact. How does this look? Proof: In order to show that $A$ is compact. We need to show that for any open ...
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82 views

Three definition of total boundness (for uniform spaces)

The following are three definitions of a totally bounded uniform spaces on a set $U$: For every entourage $E$ there exists a finite cover $S$ of $U$ such that $\forall A\in S:A\times A\subseteq E$. ...
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53 views

Showing continuity of a function that depends on another continuous function.

Question: please help me pointing out the errors of my proof (I'm sure there are some). The proof is structured in cases (two cases with each two subcases) and I think that some may be correct but ...
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1answer
18 views

About the boundary of a set of the form $Q_i = \bigcup_{t \in (0,T)}\Omega_i(t) \times \{t\}$

Let $\Omega$ be a bounded (open) domain. For every $t \in [0,T]$, let $\Omega_1(t), \Omega_2(t)$ be open subsets of $\Omega$, with $S(t)$ the interface separating $\Omega_1(t)$ and $\Omega_2(t)$. ...
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79 views

Topologies on Power Sets

My assignment is the following: Let $X$ be a topological space with topology $T$. Define \begin{align*} Q =\{ A \subset P(X) \mid \bigcup_{a \in A} a\in T\} \\ R = \{ A \subset P(X) \mid ...
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73 views

Open Mapping: What good for?

Good morning everybody, I'm wondering what open mappings are actually good for (except for inverse becomes continuous)??? My irritation came since, people stress that an open mapping not necessarily ...
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24 views

$\forall C \subseteq A$, $\exists C' \subseteq A : C \subseteq D \Rightarrow C' \subseteq D'$ then $\exists E \subseteq A : E = E'$

Theorem: Suppose that with each subset $C$ of $A$ there is associated a subset $C'$ of $A$ in such a way that $C \subseteq D$ implies $C' \subseteq D'$. Then $E = E'$ for some $E \subseteq A$. ...
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0answers
118 views

Hatcher 0.5 without the Tube Lemma?

Can Hatcher 0.5 be proven this way? This was my first proof before I read the posted version that cites the Tube Lemma. Let $F(t,x): I \times X \rightarrow X,$ such that $F(0,x) = f_0(x) = 1_X$, ...
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14 views

Triangle Mesh as a topological disk [duplicate]

I was reading up on the Dirichlet problem, and was truly hoping if anyone here has the time to help make me understand this a bit better. In particular, the question relates to harmonic maps. My ...
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1answer
43 views

Question about the proof of $S^3/\mathbb{Z}_2 \cong SO(3)$

I'm trying to show $S^3/\mathbb{Z}_2 \cong SO(3)$ completely rigorously. For that purpose I considered three-sphere $S^3$ as a subspace of the ring of quaternions $\mathbb{H}$ and looked into the map ...
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51 views

Orbits of discrete groups

Notice that isometry group of euclidean space$\Bbb{R}^n$ is displayed by E(n). I would like know that why any discrete subgroup G of E(n) ( i.e subspace topology (from E(n)) on G is discrete) has ...
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102 views

Discrete subgroups of isometry group $\mathbb{R}^n$

Let $G$ be a Hausdorff topological group. We say that a subgroup $S$ of $G$ is discrete if and only if the subspace topology (from $G$) on $S$ is discrete. Note that isometry group of euclidean space ...
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92 views

When is a Lipschitz homeomorphism of metric spaces bi-Lipschitz?

Let $(X,d_X)$ and $(Y, d_Y)$ be metric spaces, and let $f: X \to Y$ be a Lipschitz map which is a homeomorphism of the underlying topological spaces. Are there conditions which assure that $f$ is ...
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63 views

Proof a function is continous.

Is the function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, where $f(x_1,x_2) = x_1^2 + x_2^2$, a continuous function? My attempt: Suppose that $\forall \varepsilon > 0$ $\exists \delta >0$ such ...
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1answer
106 views

Hausdorff Measure and Hausdorff Dimension

Could someone explain the intuition behund the Hausdorff Measure and Hausdorff Dimension? The Hausdorff Measure is defined as the following: Let $(X,d)$ be a metric space. $\forall S \subset X$, ...
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1answer
127 views

If the distance between any two points is less than $1$, must $X$ be compact?

Let $X$ be a complete metric space such that the distance between any two points is less than $1$. Then is $X$ necessarily compact? Thanks in advance.
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198 views

If we draw infinitely many lines on a table, can we find a triangle somewhere? [closed]

If we draw infinitely many lines on a table, can we find a triangle somewhere? We prove that there is a subgraph $C_3$ in $C_n$, which will be called a triangle. Suppose we have an infinite ...
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236 views

$f,g$ continuous from $X$ to $Y$. if they are agree on a dense set $A$ of $X$ then they agree on $X$

$$ \textbf{PROBLEM} $$ Suppose $f$ and $g$ are two continuous functions such that $f: X \to Y $ and $g : X \to Y $. $Y$ is a a hausdorff space. Suppose $f(x) = g(x) $ for all $x \in A ...
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1answer
47 views

Convergence/Sequences/Box Topology

Let $\mathbb{R}^\omega$ be the countable product of $\mathbb{R}$. Make it a topological space using the box topology. Show that the sequence $\{(1/n, 1/n, ....)$ | $n \in \mathbb{Z}_+\}$ does not ...
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265 views

find closure, interior, and derived sets with respect to topologies

I have a midterm coming up and I'm trying to understand this problem. I understand what closure and interior mean but the different topologies are a little confusing to me. Let $\mathbb{R}$ be the ...
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51 views

contractive property

A topological property $\mathcal{R}$ is contractive if $(X,\tau)$ has property $\mathcal{R}$ and if $\tau^{\prime} \subset \tau$, then (A, $\tau^{\prime}$) has property $\mathcal{R}$. A topological ...
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33 views

Question on sequences in Upper limit topology of reals [duplicate]

2) Consider the upper limit topology: $N(x)=\lbrace N\subseteq \mathbb{R}∣ \exists a,b \in \mathbb{R}:x \in (a,b]\subseteq N\rbrace$ A) Prove that N is a neighborhood topology on $\mathbb{R}$ B) ...
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75 views

Convergence of sequence in uniform and box topologies

I am trying the following problem: $w_1=(1,1,1,1,\ldots)$ $w_2=(0,2,2,\ldots)$ $w_3=(0,0,3,3,\ldots)$ $\cdots$ $x_1=(1,1,1,1,\ldots)$ $x_2=(0,\frac{1}{2},\frac{1}{2},\frac{1}{2}\ldots)$ ...
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50 views

“It can be checked locally that $Z$ is a closed subset”

Look at the following proposition/exercise: A subspace $Z$ of a topological space $X$ is closed if and only if exists an open cover $\{U_\alpha\}$ of $X$ such that $Z\cap U_\alpha$ is closed in ...
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1answer
100 views

Is a set of bounded functions bounded?

Please consider the following question (note that $C_b$ is the space of bounded continuous functions): Let $f_k$ be a convergent sequence in $\mathscr C_b(A, \mathbb R^m)$. Prove $\{f_k \mid k = ...
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143 views

If S is a closed set, prove that $\partial S$=$\partial(\partial S)$

If S is a closed set, prove that $\partial S$=$\partial(\partial S)$. I'm trying to prove this using the equation $\partial S$=cl($\partial S$)=int($\partial S$)$\cup \partial(\partial S)$, then we ...
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1answer
78 views

free and uniform ultrafilter

An ultrafilter $\mathcal{F}$ is said to be free if $\cap \mathcal{F} = \emptyset$. An ultrafilter $\mathcal{F}$ is an uniform ultrafilter in $X$ if $|F| = |X|$ for every $F \in \mathcal{F}$. ...
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99 views

How can I prove these statements?

I would be very glad if you helped me solving this problem. It's about topology. Okay let's begin : Let $A$ be a non-empty open subset of $\mathbb R$. Let $x_0 \in A$ and $B = [x_0,\infty) ∩ A^c$ ...
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121 views

How can I prove that the closure of a set in a metric space is the set of all limits of sequences in that set?

Let $(X,d)$ be a metric space and let $A \subsetneqq X$. Let $E$ be the set of all $p$ in $X$ for which there exists a sequence $(p_n)$ in $A$ such that $ p = \lim\limits_{n\to \infty}p_n$. Show that ...
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1answer
47 views

Under what conditions $f$ and $g$ are open maps?

Let the composition of two arbitrary function $f$ and $g$ be an open function. Under what conditions $f$ and $g$ are open maps?
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39 views

How to prove that every uniform space is completely regular?

Wikipedia says "every uniform space is completely regular". How to prove that every uniform space is completely regular?
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42 views

Posets and Level sets

We consider the cube $L=[0,r_0]^3$ where $r_0 > 0$ is a fixed real number. The lattice is generated by the level set $L_r$ We define the level function $\lambda (x,y,z) \to x+y+z =r$ For $ r ...
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111 views

If product of two sets $A\times B$ is closed, are $A$ and $B$ closed?

If $A\times B$ is closed in $X\times Y$, then are $A$ and $B$ closed in $X$ and $Y$ respectively?
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1answer
57 views

Each countable Hausdorff space is Katetov KC

A space $ ( X, \tau )$ is said to be Katetov- KC if there is a topoloy $\sigma \subset \tau$ such that $(X,\sigma) $ is minimal KC. Theorem: Each countable Hausdorff space is Katetov KC. proof: ...
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127 views

If $A$ is compact and $B$ is Lindelöf space , will be $A \cup B$ Lindelöf

I have 2 different questions: As we know a space Y is Lindelöf if each open covering contains a countable subcovering. (1) :If $A$ is compact and $B$ is Lindelöf space , will be $A \cup B$ ...
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66 views

infinite regular cardinal

Let $(X,\tau)$ be a KC non-compact space. Then there is a discrete subset $D \subseteq X$ such that $\overline D$ is not compact. Furthermore there is an ultrafilter $F$ in $X$ such that $ D \in F $ ...
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1answer
36 views

Let $X$ be a$KC$ space. Then $X^{*}$ is $KC$ iff $X$ is a $K$ -space

The bellow theorem exist in " Between $T_{1} $ and $T_{2}$ " by " Albert Wilansky ." Let $ (‎ ‎X‎^{*},‎\tau‎^{*} ‎)‎ $ be topological space one- point compatification of $ ( X. \tau)$. A topological ...
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37 views

sequential $US$-space

A topological space is called a US-space provided that each convergent sequence has a unique limit. the notion of strongly KC-spaces, that is, those spaces in which every countably compact subset is ...
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64 views

Definition of initial segment

the definition of initial segment is correct: "let $\preceq$ be a linear ordering of a set $A$, and $B \subsetneqq A$, $B$ is initial segment of $A$ under $\preceq$ if $\forall a \in A, \forall b \in ...
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59 views

minimal KC and (strongly) KC

If P is a topological property, then a space (X, τ) is said to be minimal P (respectively, maximal) if (X,τ) has property P but no topology on X which is strictly smaller (respectively, strictly ...
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86 views

topological space

Let $( X,\tau )$ be a $T_1$ topological space. Let $D = \{ d_n : n \in \omega \}$ be a countably infinite closed discrete subspace of $X$. Fix $P \in X$ and let $F \in \beta\omega- \omega$ be an ...
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1answer
91 views

How to prove that any ball minus a point is a path-connected set?

Let $B_1\subset\mathbb{R}^n$ be a open ball and $B_2\subset\mathbb{R}^n$ be a closed ball, where $n\geq2$. How to show that if $p_1\in B_1$ and $p_2\in B_2$, then $B_1\setminus \{p_1\}$ and ...
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144 views

An exercise about countable basis

In the book Munkres: Topology §30 I met the following problem: Show that if $X$ has a countable basis $\{B_n\}_{n \in \mathbb{Z}_+}$, then every basis $\mathscr{C}$ for $X$ contains a countable ...
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1answer
110 views

prove that if $V$ is open in $\Bbb R^n$ then there are open balls such that $V=\bigcup_{j\in\Bbb N} B_j$

Prove that if $V$ is open in $\Bbb R^n$ then there are open balls such that $V=\bigcup_{j\in\Bbb N}B_j$. I have the solution, but it is too short and it is not enough to prove it, also it's too ...
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2answers
186 views

If $f$ is continuous then $G$ is homeomorphic to $X$.

Let $f: X \to Y$ be a function. The graph of $f$ is defined to be the set $G = \{(x, f(x)) : x \in X\}$. Prove that if $f$ is continuous then $G$ is homeomorphic to $X$. an anyone suggest me the ...
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1answer
93 views

If a space $Y$ is a countable union of closed discrete subspaces, then is $Y$ a $\sigma$-space?

A space $X$ is a $\sigma$-space if $X$ has a $\sigma$-discrete network. If a space $Y$ is a countable union of closed discrete subspaces, then is $Y$ a $\sigma$-space? I think it is. But I'm not ...
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79 views

Is $\mathbb{R}^2$ boundaryless?

I just have a quick question, as stated in the title. Is $\mathbb{R}^2$ boundary-less? Thank you very much. :-)
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1answer
88 views

Let $X$ be a Moore space and $e(X)=\omega$. Is it metrizable?

Let $X$ be a Moore space and $e(X)=\omega$. Is it metrizable? What I've tried: I list these facts: 1 A space $X$ is a Moore space iff $X$ is a $\sigma$-space and a $p$-space. 2 If $X$ is a ...
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1answer
74 views

Continuous function between a compact set and closed set for a Tychonoff space

Please help me in proving this theorem. If $A$ is a compact subset of a Tychonoff space $X$, then for every closed set $B \subset X \setminus A$, there exists a continuous function $f:X \to I$ ...
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2answers
70 views

$A_n$ is decreasing to a singleton. Do all neighborhoods of the singleton, contain almost all $A_n$?

Let $(X,\mathcal T)$ be a locally compact Hausdorff homogeneous topological space and $(A_n)$ be a decreasing sequence of subsets of $X$ and $$\bigcap_{n}A_n=\{a\}$$ be a singleton. Is there some $n$ ...