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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25 views

Open set in the sup-metric space

can you please explain why is the following so? Or at least point me in a direction which can help me find the answer? Given a set G of functions g: $\mathbb{R} \rightarrow \mathbb{R}$ such that ...
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1answer
44 views

Vector bundle over a compact, Hausdorff space is a summand of a trivial bundle.

I am trying understand the proof of the following (proposition 1.4 in Hatcher's book on Vector Bundle). For every vector bundle $E\overset{p}{\to} B$, with $B$ compact Hausdorff, there exists a ...
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0answers
40 views

A Question on a second countable $T_2$ space

I have a question: Does a second countable $T_2$ space $X$ always have a $G_\delta$-diagonal? (If $X$ is regular, then it is metrizable, and it obviously has a $G_\delta$-diagonal.) Thanks for your ...
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3answers
119 views

Is the boundary of a boundary a subset of the boundary?

The definition of a boundary of a set $S$ in a topological space $X$ is $\text{comp}\{\text{Int}(S) \cup \text{Ext}(S)\}$ (complement of the interior union exterior). The definition for interior is ...
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2answers
49 views

Is $\mathcal{T'} = \{S : S \subset X, S^C \in \mathcal{T}\}$ a topology on $X$?

If $\mathcal{T}$ is a topology on $X$, then is $\mathcal{T'} = \{S : S \subset X, S^C \in \mathcal{T}\}$ a topology on $X$? Since $S^C \in \mathcal{T}$, then $S$ is a closed set of $X$, if $X$ is ...
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0answers
58 views

sequential continuity and countinuity

When we have two topological spaces, $\left(X, \tau_X\right)$ and $(Y, \tau_Y)$ it is easy to check that for $f: X \rightarrow Y$ continuity implies sequential continuity. I'm wondering what do we ...
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2answers
39 views

Order and Topology

Given any set, a total order on that set can induce a topology on that same set. Does the opposite also work ? Given a topology on a set, can it induce an order ( perharps total ) on that set ? ...
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1answer
63 views

What is it homeomorphic to?

Today a friend of mine did me the following question. Could you help me to understand what is it homeomorphic to? Considering in $\mathbb{E^2}$ the following topologycal subspace: $$X=\{(x,y) \in ...
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1answer
55 views

Can the ball $B(0,r_0)$ be covered with a finite number of balls of radius $<r_0$

Consider an infinite dimensional Banach space $X$. Let $B(0,r_0)$ be the ball with radius $r_0$. We know that the ball $B(0,r_0)$ is not relatively compact, so it is not totally bounded. This implies ...
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1answer
23 views

How do the interval and lower topologies relate when generated by an arbitrary poset?

At first glance, it would seem that the lower topology (and the upper topology, for that matter) would be a subset of the interval topology for a partially ordered set P, since the open-ended ...
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1answer
44 views

Show any open interval is a half open set?

How do I show that any open interval is an half open set and use this to conclude that any open set is also half open? I am in an introduction to proofs writing class. I have a feeling I need to use ...
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48 views

Prob. 8, Sec. 19 in Munkres' TOPOLOGY, 2nd ed: What is the situation in the box topology?

Let $\mathbb{R}^\omega$ denote the set of all sequences of real numbers, and let $(a_1, a_2, a_3, \ldots ), (b_1, b_2, b_3, \ldots) \in \mathbb{R}^\omega$ be fixed with $a_i > 0$ for all $i= 1, 2, ...
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2answers
51 views

Every Cauchy Sequence in the reals converges

Prove that every Cauchy sequence in $\mathbb{R}$ converges. proof: Let $A_n$ be a closed and bounded sequence. Then there exists an interval $[a_1,b_1]$ such that $a_1\leq A_n \leq b_1$ $\forall n$. ...
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0answers
64 views

Limit points in nonstandard analysis [solved]

Let $A\subseteq\mathbb{R}$, $p\in\mathbb{R}$. I proved that the following are equivalent: $\exists\left(x_{n}\right)_{n\in\mathbb{N}}\subseteq A\cap\left\{ p\right\} ^{c}$ such that ...
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0answers
49 views

help in proving equivelant statements of $f=\sum\limits_{i=1}^\infty \langle f,\phi_i \rangle \phi_i \space \forall f\in H$

Let $H$ is a hilbert space. $\{\phi_i\}_{i=1}^\infty$ is an orthonormal set A set $\{\phi_i\}_{i=1}^\infty$ is complete in $H$ if any of the following statements hold: $f=\sum\limits_{i=1}^\infty ...
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0answers
109 views

$A \subset \Bbb R$ such that $A$, $clA$, $int(A)$, $cl(int(A))$, $int(clA)$ are pairwise distinct

Do there exist subsets with internal closures $A$ of $\mathbb R$ such that $A$ , $\bar A$ , $A^\circ$ , $(\bar A)^\circ$ , $\overline{A^\circ}$ are pairwise distinct? I found an example from a book ...
2
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1answer
72 views

Are open sets in $R^n$ homeomorphic to $R^n$?

I am working on exercise 1.1 and I think the way to do this would be to show that open sets are homeomorphic to $R^n$ or open balls in $R^n$. Is this even true? I'm not sure how to go about ...
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1answer
24 views

A problem probably related to Lebesgue Number Lemma

This is an exam question that i am having trouble with. I tried to show that $f$ is a closed map, but without the assumption of surjectivity i could not prove my claim of closedness. Also i wanted ...
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1answer
19 views

Let X be a nonempty set. Let x∈X. Show that the collection 𝔗={U⊆X:U=∅ or x∈U} is a topology for X.

Let $X$ be a nonempty set. Let $x \in X$. Show that the collection $ \mathfrak T = \{ U \subseteq X : U = \emptyset$ or $ x \in U \}$ is a topology for X. I know I need to show that this ...
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0answers
19 views

$A\subseteq B(X, Y)$ compact if and only if closed and $Ax$ is conditionally compact

This comes from Exercise 2 of Chapter VI in Dunford & Schwartz. I am trying to prove the following statement: A set $A\subseteq \mathscr{B}(X, Y)$ is compact in the strong operator topology if ...
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1answer
21 views

Are these topologies in $\mathbb R$ am I on the right track?

I have just been introduced to what a topology is. I understand that there are 3 requirements that must be met. My definition is $\mathbb R \in \mathfrak T$ and $ \emptyset \in \mathfrak T$ If $ ...
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1answer
19 views

Smallest integer $N(\epsilon)$ such that $K\subset \bigcup_{n=1}^{N(\epsilon)}B(x_i,\epsilon)$

In a metric space, a set $K$ is said to be totally bounded if for each $\epsilon>0$ there exist a finite number of balls $B_1,B_2\dots B_{N(\epsilon)}$ with radius $\epsilon$ which covers $K$. ...
2
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1answer
133 views

Closure of a set in a “Topology of finite complement”

Well, I was reading this article by Kelley and when reached the point where he say that $X_a$ is closed in $Y_a$ I had to stop, probably mine is just a stupid misunderstand but can't figure out how to ...
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2answers
38 views

Assuming that the graph of a function..Show that it is not continuous by finding an open set $V$ such that $f^{-1}(V)$ is not open.

Assuming that the graph about is the graph of a function $f: \mathbf R \rightarrow \mathbf R$ Show that it is not continuous by finding an open set $V$ such that $f^{-1}(V)$ is not open. I am not ...
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1answer
40 views

Showing an uncountable set is equivalent to the positive set of integers

Show that the set $\prod_{i = 1}^{\infty}\left\{1,0\right\}$ is equivalent to the set of subsets of $\mathbb{Z}^{+}$, and explain why this shows it is uncountable. Proof: We want a function that is a ...
2
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0answers
61 views

Counter-example to exponential law for locally compact [non-Hausdorff] spaces

There is a natural bijection $$ \operatorname{Map}(X\times Y,Z)\cong\operatorname{Map}(X,\operatorname{Map}(Y,Z)),\quad f\mapsto(x\mapsto(f(x,{-})). $$ If $X$, $Y$, $Z$ are topological spaces one can ...
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49 views

A game may related to graph theory or topology

Last Sunday, I played a game with a group of people. The game is as follows: A group of people form a circle as shown below: Each person must remember how he/she is linked with his two neighbours. ...
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1answer
33 views

Meager Set in Function Space (Constructing Sequence of Functions)

Let $X = \{f: [0,1] \to \mathbb{R} \; | \; f\in C^1[0,1], f \textrm{ strictly increasing} \}$ equipped with the topology of uniform convergence. Consider the subset $A =\{ f \in X \; | \; ...
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2answers
58 views

How do I show $SO(n)$ is open and closed in $O(n)$?

As title, how do I show $SO(n)$ is open and closed subset of $O(n)$? Is the preimage of closed set under continuous map closed ?
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1answer
38 views

An example for a topological space $X$ in which $C_p(X)$ is countably tight but not sequential

Is there an example for a topological space $X$ in which $C_p(X)$ is countably tight but not sequential? $C_p(X)$ is the space of continuous functions from $X$ to $\mathbb R$ with the topology ...
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1answer
29 views

How do I find the volume cut off of the unit ball by the plane ax+by+cz = d?

How do I find the volume cut off of the unit ball by the plane ax+by+cz = d? I know that there's a double integral somewhere here, but I just don't understand how to attack this problem. Any guidance ...
2
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2answers
49 views

Maximum and Minimum of Totally Ordered Compact Sets

Let T be a totally ordered compact set. Does this always imply that a maximum and a minimum element of T exist under this total order? And if not, what about the special case where: T is a closed ...
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90 views

Test whether the following sets are connected or not.

Which of the following sets are connected ? (A) $\{(x,y)\in \mathbb R^2:x,y\in \Bbb Q\}\subset \Bbb R^2$. (B) $\{(x,y)\in \Bbb R^2:\text{ at least one of } x,y \text{ is rational ...
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2answers
43 views

Let $\{p_1, . . . , p_l\}$ be points in $\mathbb{R}^n$ . Show that the set $U = \mathbb{R}^n\setminus \{p_1, p_2, . . . , p_l\}$ is open.

Question is as stated in the title. I'm aware that $\mathbb R^n$ is an open (or closed) set, and I know how to prove it, but beyond that I'm stuck. Firstly, how do I prove that the set of arbitrary ...
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1answer
42 views

Proof that $A$ and $B$ nonempty and closed in $\Bbb C$ means that $A\cup B$ is closed in $\Bbb C$ verification

Is the below proof rigorous? Let $A$ and $B$ be nonempty and closed in $\Bbb C$. Assume $A\cup B$ is not open, then there is a limit point $p$, not in $A\cup B$, that must be in $\overline{A}$ ...
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66 views

Let $A \subset X$ and $Y = (X-A) \cup \{A\}$ with $(X,\tau_x)$ a topological space

Let $A \subset X$ and $Y = (X-A) \cup \{A\}$ with $(X,\tau_x)$ a topological space. now define $\pi: X \rightarrow Y$ by $\pi(x) = x$ if $x\notin A$ and $\pi(x) = \{A\}$ if $x\in A$ consider $Y$ with ...
3
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1answer
44 views

Is every second countable $T_2$ topological space a developable space?

I have a question: Is every second countable $T_2$ topological space a developable space? Thanks for your help.
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0answers
35 views

A map of two Euclidean spaces preserving connectedness and compactness is continuous [duplicate]

Let $ f:\mathbb R^n \to \mathbb R^m$. If $f$ preserves connectedness and compactness then $f$ is continuous. How can this be proven? I don't really know where to start.
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1answer
40 views

Is it path-connected space?

I have a finite topological space $X= \{ 1,...n \}$ with the following topology that I created $T=\{ \emptyset, X \} \bigcup \{A \subseteq X | 1\in A\} $ It is connected because $\bigcap_{A \in X, ...
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60 views

$f(z)=\frac{1}{z}$ has an antiderivative on any simply connected domain

Prove that the function $f(z)=\frac{1}{z}$ has an antiderivative on any simply connected domain of $\Bbb C$ which does not contain zero. Also prove that this function does not have an ...
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1answer
30 views

Mackey topology on $X'$

Does the Mackey topology $\tau(X',X)$ coincide with the operator norm topology on the dual $X'$ of a normed space $X$?
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1answer
38 views

The image of a segment is not dense into a square

Let $f\colon [0,1]^2 \to \mathbf{R}^2$ a continuous injection such that $$ f(0,0)=(0,0), f(1,0)=(0,1), f(1,1)=(0,2), f(0,1)=(0,3). $$ Let $S$ be the closed segment joining $(0,0)$ and $(0,3)$. Prove ...
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0answers
50 views

$F \subset X= \prod_{\alpha}x_\alpha \ \alpha \in I$ is closed if and only if $F$ is the finite intersection of…

I'd just like to see if the following proof is reasonable - and if not, what went wrong. Thx for taking a look! $F \subset X= \prod_{\alpha}x_\alpha \ \alpha \in I$ is closed if and only if $F$ is ...
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1answer
34 views

Proving that $\Omega$ is bounded.

$\Omega$ is a non-empty subset of $\mathbb{C}$ such that every sequence of $\Omega$ has a subsequence which converges to some point in $\Omega$. I need to prove that $\Omega$ is compact( a set $S$ is ...
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1answer
44 views

Topology, Showing that two metric spaces are topologically equivalent

Can someone verify if this is true? $X=\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ . $d$ is the standard metric on $\mathbb{R}$ and $d'(x,y)=d\left(\tan(x),\tan(y)\right)$ . We want to show that ...
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1answer
46 views

Countinuity of the identity map between different topologies

Could you please verify this (rather simple) proof? I'm a bit new to this kind of reasoning. The question is: Let $\mathcal{T}_1$, $\mathcal{T}_2$ be two topologies on some set $X$, when is the ...
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2answers
209 views

Orbits of properly discontinuous actions

Definition Let $G$ be a group and $X$ a topological space. Let $G\curvearrowright X$ by homeomorphisms. We call the action properly discontinuous if for all $x\in X$ there exists an open neighborhood ...
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1answer
32 views

wot limit of a sequence of projections

Let $\{P_i\}$ be a net of projections on a Hilbert space , then we can show wot limit of this net is a projection, too. I saw below example of a sequence of projections which its wot limit is not a ...
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1answer
52 views

Weak topology generated equal to the original topology

If we have ($\mathbb{R}$, standard topology), I was wondering what are the conditions that are given to the family of functions will result in us having a topology equal to it's original self. I was ...
1
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1answer
25 views

Two definitions of attaching are equivalent.

Suppose $X,Y$ are topological spaces, $A\subset X$ and $$ f: A\to Y $$ is a quotient map, that is, a surjective continuous map with $U\subset Y$ open if and only if $f^{-1}(U)\subset A$ is open. We ...