0
votes
1answer
48 views

Correct proof of supremum property?

Let $u$ be an upper bound of non-empty set $A$ in $\mathbb{R}$. Prove that $u$ is the supremum of $A$ if and only if for all $\epsilon > 0$ there is an $a \in A$ such that $u-\epsilon < a$. ...
0
votes
0answers
30 views

The set of rational numbers, each point is point accumulation

Please let us help someone by telling you a precise formulation is below, and then someone please tell me solution that has since become like that with a few days my friend we debates, here my ...
0
votes
0answers
19 views

Question about Boundary points of the sets in metric space

Let A be a metric spaces. Prove the following properties: The boundary of $A$ equals $A'-A$ The boundary of $A$ is the closed set. $A$ is closed if and only if it contains its boundary. Where ...
0
votes
1answer
44 views

Minkowski Distance Metric

Given compact sets $A$, $B$, define the Minkowski distance between the two sets as: $$ \delta(A,B):= \inf \{ r: B \subseteq \mathscr{N}_r (A) \, \, \text{and} \, \, A \subseteq \mathscr{N}_r (B) \}$$ ...
2
votes
2answers
34 views

Topology and Arithmetic Progressions

I'm self-studying from "Elementary Topology Problem Textbook" by O.Ya.Viro et al. This is Exercise 2.Lx : Consider the following property of a subset $F$ of the set $\mathbb{N}$ of positive ...
0
votes
1answer
29 views

Can we deduce that $X$ is $\sigma-$compact? [on hold]

Assume that a quotient space of the space $X$ is compact. Can we deduce that $X$ is $\sigma-$compact?
0
votes
1answer
25 views

Question regarding a wording of an exercise related to Noetherian topological space

The exercise states "If $X$ is a Noetherian topological space, show that the union of any subset of the connected components of $X$ is always open and closed in $X$." Does the question mean "If I ...
1
vote
0answers
61 views

Homeomorphism between positive rationals and non-negative rationals [duplicate]

Professor told me the two spaces are homeomorphic if we consider them as subspaces of R equipped with the standard topology. It's a bit surprising for me since I have already found Q is homeomorphic ...
0
votes
1answer
56 views

Density and convergence

I have a small question: Is it true that if the basis of a space $A$ is dense in a space $B$ ($B\subset A$) then if $u_n\rightarrow u$ in $A$ we have that $u_n\rightarrow u$ in $B$ ?
2
votes
1answer
52 views

For compact $K$ and open $U \supseteq K$, there exists $\varepsilon>0$ such that $B(K,\varepsilon) \subseteq U$

Let $X$ be a metric space. Let $K$ be a compact subset of $X$ and $U$ an open subset of $X$ containing $K$. I strongly believe and want to prove that there exists $\varepsilon>0$ such that ...
1
vote
1answer
48 views

Deck transformation acting properly discontinuously assumed covering space is path-connected

Let $p:E\rightarrow X$ be a covering space, $x\in X$ fixed point, $E$ path-connected and $\Delta(p)$ – Deck transformation group of $p$, that is $\Delta(p) = \{f\in \text{Homeo}(E):pf=p\}$. Let ...
4
votes
1answer
62 views

Is continuous map from covering space to itself homeomorphism assumed both cover and base path-connected and $pf=p$?

In my topology assignment I came across the following problem: True or false? Let $E$ and $X$ be path-connected. For every covering map $p:E\rightarrow X$ and continuous map $f:E\rightarrow E$ ...
1
vote
1answer
58 views

Compact $\omega$-limit set $\Rightarrow$ connected

Consider the flow $\varphi: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ and $L_{\omega}(x)$ the $\omega$-limit set of a point $x \in \mathbb{R}^n$. How can I show that if $L_{\omega}(x)$ is ...
3
votes
1answer
29 views

Show that $Y$ is not path-connected

Let $\mathbb{R}^2$ with the usual topology and let $$ Y = A_0 \cup (\bigcup_{n \in \mathbb{N}} A_n) \cup (\bigcup_{n \in \mathbb{N}}L_n)$$ where $$ A_0 = \{ 0 \} \times [0,1] \qquad A_n = \{ ...
0
votes
1answer
39 views

a topological space is the union of its irreducible components

if we define irreducible component of a topological space $X$ as the maximal closed irreducible subset of $X$,prove that we can write $X$ as the union of its irreducible components. how to approach ...
-1
votes
1answer
40 views

a quotient space homeomorphic with $\mathbb{R}\mathbb{P^2}$

prove that if we glue the closed unit disc to the circular boundary of the mobius strip we obtain a quotient space that is homeomorphic with $\mathbb{R}\mathbb{P^2}$. it is my general topology ...
1
vote
1answer
50 views

$\mathbb{R}\mathbb{P^2}$ as a quotient space

If we construct $\mathbb{R}\mathbb{P^2}$ by gluing the sides of $I^2$, can anyone explain for me that why $\mathbb{R}\mathbb{P^2}$ can be considered as the quotient space induced by the equivalence ...
0
votes
0answers
44 views

one point compactification of upper half plane

"prove that the closed unit disc could be considered as the one point compactification of the upper half plane with the X-axis as the bounday" i believe in this statement by the imagination,but i ...
0
votes
1answer
18 views

Let $K_1(0) := \{x \in \mathbb{R}^2: \|x\|_2 < 1\}$ and $S := K_1(0) \setminus \mathbb{Q}^2$. Is M path connected?

The Assignment: Let $K_1(0) := \{x \in \mathbb{R}^2: \|x\|_2 < 1\}$ and $S := K_1(0) \setminus \mathbb{Q}^2$. Is S path connected? Explain your answer. I don't think S is path-connected since ...
0
votes
2answers
26 views

Question about convex set?

Can someone prove or give me a counterexample that is for a set $A$, if the closure of $A$ is convex then the interior of $A$ is also convex?
2
votes
2answers
100 views

$f:X\rightarrow S^1$ a continuous map. $X$ a path-connected topological space.

Let $f:X\rightarrow S^1$ be a continuous map from a path-connected topological space $X$ and let $p:\mathbb{R} \rightarrow S^1$ be the universal covering. What is the condition when $f$ admits a ...
2
votes
1answer
28 views

Topology - A continuous map from the circle that can be extended to a continuous map from the disc is null

a question from my h.w. Let $h:S^1 \rightarrow X$ be continuous map, and suppose it can be extended continuously to the Disc meaning there exists a continuous map $k:D^2 \rightarrow X$ such that ...
2
votes
0answers
59 views

Solution check please: Basic topology (open/closed sets and balls)

This is a part of a problem from a past exam so i need my solution to be reviewed. Thank you in advance for taking your time :) Let $\Omega$ be a bounded open set and let $A= \{r>0| \exists z \in ...
0
votes
1answer
57 views

Subset of infinite connected set

How to proove that infinite connected set has got proper infinite connected subset?
4
votes
2answers
77 views

(Certain) colimit and product in category of topological spaces

Consider the diagram $$(*):\;\;\;X_0 \stackrel{i_0}\hookrightarrow X_1 \stackrel{i_1}\hookrightarrow X_2 \stackrel{i_2}\hookrightarrow \cdots $$ in category of topological spaces. Denote $I$ the ...
1
vote
0answers
26 views

Principal order filters on a POSET

I have another problem, but in this one I have no idea how to start. Let be $(X,\leq)$ a POSET with a first element and gifted with the topology $\{ (a,\rightarrow) : a \in X \}$ (principal order ...
2
votes
0answers
52 views

isomorphism of algebric torus

I'm trying to prove the following: Let $D_n=(\mathbb{C}^{\times})^n$ (an algebric torus of rank $n$). Assuming $D_k$ is isomorphic to $D_n$ as an algebric group. Prove that $k=n$. So far, I managed ...
4
votes
1answer
42 views

Compactness and sequential compactness in metric spaces

I got a question: I'm trying to proof that every metric space is compact if and only if the space is sequentially compact. In all the proves I have found, they used the Bolzano-Weierstrass theorem. Is ...
0
votes
1answer
55 views

Question about Alternating forms

So I understand the definition of an alternating form on $\mathbb{R}^m$, but I don't really understand the proof of the lemma. Could someone explain the first observation? Why is it so?
0
votes
1answer
41 views

How to construct/describe the set of unions of all intervals $(a, b)$ of the real line?

I have this problem in my textbook where I should verify whether a set $\Omega$ is a topological structure in $X=\mathbb R$. $\Omega$ is said to be "the set of unions of all intervals $(a, b): a, b ...
1
vote
1answer
46 views

Prove that $\overline{Int\,(\overline{Int\,A})} = \overline{Int\,A}$

I need to proof this statement, and I don't know where to start. In every topological sapce, we have that $\overline{Int\,(\overline{Int\,A})} = \overline{Int\,A}$ I tried to show that ...
0
votes
1answer
21 views

Proving that $b \in \overline{A}$ if and only if $\rho(b,A) = 0$

I need some help with this problem: Let be $(X,\rho)$ a metric space, $A \subseteq X$ and $b \in X$. The distance from $A$ to $b$ is defined as $\rho(b,A) = \inf\,\{ \rho(b,a) : a \in A \}$. Prove ...
1
vote
0answers
54 views

“forgetting base point”-map properties

I need help with the following problem: Let $P:\pi_1(X,x_0)\to S(X)$ be the map from the set of homotopy classes of loops based at $x_0$ to the set of homotopy classes without restriction on the ...
1
vote
4answers
67 views

Closure of union of two sets

Show that $$\overline{A\cup B} = \overline A\cup\overline B$$ $\overline A=A\cup A'$ where $A'$ are the limit points My attempt: Since $\overline A$ is closed and $\overline B$, the union of two ...
1
vote
0answers
37 views

Classic applications of Baire category theorem [duplicate]

Problem: Suppose that $f:\Re^+\to\Re^+$ is a continuous function with the following property: for all $x\in\Re^+$, the sequence $f(x), f(2x),f(3x)\cdots$ tends to $0$. Prove that ...
0
votes
0answers
42 views

Showing the bijection to prove that two sets are equivalents

I need to prove that two sets $X$ and $Y$ are equivalent by showing the bijection between them, where $X = \{x: Ax = c\}$ and $Y = \{y: Ay\le b,-Ay\le -b\}$. $A$ is a matrix, $x$,$y$,$b$ and $c$ are ...
0
votes
1answer
26 views

Show that set is path connected?

How do I show that the set $A = \{(x,y) \in R^2: x \geq 0, y \geq 0\} \cup \{(x,y) \in R^2: x \leq 0, y \leq 0\}$ is path connected. I know that I need to construct a continuous function $f:[0,1] ...
1
vote
1answer
15 views

Neighbourhood filter of an isolated point of a topological space

How can I prove this? Let $(X,\tau)$ a topological space, and $x \in X$. Then the neighbourhood filter $\mathcal{V}(x)$ is an ultrafilter if and only if $x$ is an isolated point of $X$ Thanks a ...
1
vote
1answer
35 views

Find a topological space such that $(A^a)^a \nsubseteq A^a$

I'm having some troubles with this problem, I hope you could help me: "Given $(X,\tau)$ a topological space, find $A \subseteq (X,\tau)$ such that $(A^a)^a \nsubseteq A^a$, where $A^a = \{ x \in A : ...
1
vote
1answer
33 views

A Property equivalent to being a closed map

Please, I like you to view this statement and tell if I'm doing something wrong. Proposition: A function $f:X \rightarrow Y$ between topological spaces is closed if and only if for all $A \subseteq ...
0
votes
1answer
20 views

Set of limit points on a topological spaces.

I like to solve this problem. "Problem: In any topological space, the set of limit points of a sequence is closed." The proof is easy when we work with metric spaces, but how can I generalize this ...
0
votes
1answer
42 views

The metric space containing a compact subset is separable

Let (X,d) be a metric space and K be a cpt subset of X. If it is possible to derive 'X is compact', then since compact metric space is separable, X is separable. But I'm not sure that X is compact. Do ...
0
votes
1answer
54 views

Questions on connectedness

I have a final examination in general topology this week, and I've been doing past papers for the past two days in anticipation for it. I'm not sure if my answers are correct so could someone tell me ...
1
vote
0answers
21 views

Prove that if $p: Y \to X$ is a covering space and $X$ is path connected, then the cardinality of $p^{-1}(X)$ is constant. [duplicate]

Let $p: Y \to X$ be a covering space and $X$ is connected. I want to show that $\forall x \in X$ the cardinality of $p^{-x}$ is the same. $\textbf{My Attempt:}$ Let us first fix a point $x_0 \in X$ ...
-1
votes
2answers
43 views

A Fundamental Property of Metric Spaces …

Let $(X,d)$ be a metric space and $A\subset X$ and also suppose that $G$ is open in $X$ prove the identity: $$ \overline {G\cap A}=\overline {G\cap \overline A} $$ Proposition: The intersection of ...
1
vote
1answer
40 views

Does there exist a subset $S\subset\mathbb R$ such that inf $\{a>0:S+a=\mathbb R-S\}=0$?

I founded the following question a good challenge in real analysis and topological properties of real line... Does there exist a subset $S\subset\mathbb R$ such that inf $\{a>0:S+a=\mathbb ...
-1
votes
1answer
24 views

Some Topological Properties of Starlike Sets!

A subset $E$ of $\mathbb R^n$ is starlike if it contains a point $p_0$ (called a center for $E$) such that for each $q\in E$, the segment between $p_0$ and $q$ lies in $E$. For more information please ...
3
votes
0answers
47 views

Prove that if $f:D^2\to D^2$ is a homeomorphism, then $f(S^1)=S^1$

I've already proved that set of points $z\in D^2$ such that $D^2-z$ is simply connected is precisely $S^1$. Now from this, I'm supposed to conclude that if $f:D^2\to D^2$ is a homeomorphism, then ...
0
votes
0answers
22 views

Find a circle which is a strong deformation retract of $\mathbb{R^2}-x_0$

Let $x_o\in\mathbb{R^2}$. Find a circle which is a strong deformation retract of $\mathbb{R^2}-x_0$ Proof: If $x_0=(a,b)$, Let $S=\{(x-a)^2+(y-b)^2=1\}$. Then $f_t(r,\theta) = (\exp((1-t)r),\theta)$, ...
0
votes
1answer
20 views

Prove that $h$ and $p*q$ are homotopic relative to {$0,1$}

Let $0<s<1$. Given paths $p$ and $q$ with $p(1)=q(0)$, define $h$ by the formula $$h(t) = \begin{cases} p(t/s),& \text{if} \quad 0 \leq t \leq s \\ q((t-s)/(1-s)), &\text{if} \quad ...