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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3
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2answers
316 views

Dense subset of Cantor set homeomorphic to the Baire space

Does anyone know a proof that the Cantor set, $\{0,1\}^{\mathbb{N}}$, has a dense subset homeomorphic to the Baire space, $\mathbb{N}^{\mathbb{N}}$? Thank you.
0
votes
3answers
71 views

Is there exist such $f$

Is there exist an $f:X\to \mathbb{R}$ where $\sup f(X)= +\infty$ and $f$ is uniformly continuous on $X$ where X is bounded? I think there should not be existing such $f$ as the change of the value of ...
2
votes
1answer
53 views

Quotients of $\mathbb{R}^{3}$

What quotients can I take of $\mathbb{R}^{3}$ that give me $\mathbb{R}^{3}$ back? Thankyou
1
vote
1answer
240 views

Continuity and Boundedness

I want to show that if $f$ is a uniformly continuous real valued function on the bounded set $E \subseteq \mathbb{R}$, then $f$ is bounded on $E$. I want to define an open cover of $E$, then say ...
-1
votes
1answer
42 views

Nullhomotopy of submanifolds

Let $X=\mathbb{R}^n\backslash \{A\}$ where $A$ is a $m$-dimensional submanifold in $\mathbb{R}^n$. Under what circumstances is every $l$-dimensional submanifold of $X$ nullhomotopic? (I am talking ...
5
votes
1answer
167 views

Can a path avoid a dense set?

My actual question is related to an unusual circumstance in a game which I am playing. We are trying to move one point-like object from one part of the moon to another without passing through another ...
1
vote
0answers
64 views

How to prove this isotopy exist?

Let $M$ be a topological manifold and $N$ is a subset of $M$. Let $f_t$ be an isotopy from $id $ to $f$ which is a homeomorphism on $M$. Suppose $f(N)=N$ and there is an isotopy $g_t$ on $N$ such that ...
1
vote
1answer
371 views

Is my proof correct- totally disconnected set.

Show that the Cantor set, C, is totally disconnected. Let $x,y\in C$ and suppose (WLOG) that $x<y$. Since $x,y\in C$, then $x,y\in C_n$ $\forall n\in$ℕ . Then, by construction of C, $\exists$ ...
2
votes
0answers
60 views

How to prove an isotopy relative to a point exist?

Let $M$ $ $ be a differential manifold, and $f$ a diffeomorphism on $M$ which is isotopic to $id$. Assuming that $x\in M$ is a fixed point of $f$ and the orbit of $x$ under the isotopy is a trivial ...
6
votes
4answers
5k views

Closed sets and bounded sets

Intuitively for me, it seems as if closed sets are bounded, especially considering closed sets contain all limit points. But I know this isn't the case, because $ℝ$ is closed (and open) and is not ...
1
vote
2answers
195 views

Proof: if $K\subset M$ is compact and $A\subset K$ is closed, then $A$ is compact [duplicate]

Possible Duplicate: Compactness is closed-hereditary - significance of closed property? Proposition let $(M,d)$ be a metric space where $K\subset M$ is compact and $A\subset K$ is closed. ...
0
votes
1answer
77 views

Is the every open set in a product space contains an open set with compact boundry?

Let N be an open set of YxZ. Is there exist an open set W in YxZ with compact boundry such that W is a subset of N?
0
votes
1answer
175 views

The boundary set of product space

Let $N$ be an open set of the product space $Y\times Z$ such that the boundary of $N$ is compact. Is there an open subset $U$ of $Y$ and open subset $V$ of $Z$ such that the boundary of $U\times V$ is ...
1
vote
1answer
336 views

Are all the open sets in a Euclidean space homeomorphic?

I know that open balls are homeomorphic to the entire Euclidean space, and any convex open set can be proved to be homeomorphic to the entire Euclidean space. So I was wondering if all the open sets ...
4
votes
1answer
270 views

How can one know if a set is compact?

How can one know if a set is compact? From the definition, a set is compact if for any open cover, there exist a finite subcover. However, it is not possible to list out all the open covers to a set. ...
2
votes
1answer
879 views

Is any closed ball non-compact in infinite dimensional space?

It is known that the closed unit ball $\overline{B_1(0)}$ in a normed space $X$ is compact if and only if $\dim X < \infty$. In particular, the $\overline{B_1(0)}$ is not compact if $\dim X = ...
3
votes
2answers
230 views

Span of functions dense in $L^2$

This is an exercise from Werner's Funktionalanalysis. I have to show that the linear span of the functions $f_n(x)=x^ne^{-x^2/2}, n\geq0$ is dense in $L^2(\mathbb{R})$. The book gives the hint to ...
0
votes
1answer
201 views

What are useful properties of limit inferior/superior of real-valued function?

To make it clear, this is the definition from wikipedia; Let $X,Y$ be topological spaces and $E\subset X$. Let $Y$ be an ordered set and $f:E\rightarrow Y$ be a function. Then, $\limsup_{x\to a} ...
4
votes
2answers
227 views

$c_0$ is not compact in $\ell^\infty$

Let $c_0$ be the sequences with $\lim_{n\rightarrow \infty} = 0$. Show that the closed unit ball $\{x\in c_0, \|x\| \leq 1\}$ is not compact in $\ell^\infty$. I know a lemma that says that the ...
0
votes
0answers
69 views

How do I prove that $(c,\infty]$ is open in $\overline{\mathbb{R}}$?

$\overline{\mathbb{R}}$ is a topological space, but not a metric space, so I'm not sure if it is true. Let $U$ be an open set in $\overline{\mathbb{R}}$ such that $\infty\in U$. Then, how do I that ...
2
votes
1answer
90 views

Meaning of “sequentially compact”

What does sequentially compact mean?
2
votes
1answer
24 views

Noncontinuity and an induced equivalence relation

Can someone give me an example of a map which is not continuous such that $f(\{a\}) = f(\{b \})$ induces an equivalence relation $ \{ a \} \sim \{ b \} $?
1
vote
2answers
120 views

Find two subsets $A$, $B$ that satisfy intersection-closure relations.

Problem: Let $(M,d)$ be a metric space where $A,B\subset M$. If $\overline X$ is the closure of $X$, find two sets $A$, $B$ that satisfy $$ \overline A \cap B,\;\; A \cap \overline{B},\;\; ...
1
vote
0answers
59 views

Existence of limit inferior and exterior

First of all, i'm sorry that i don't know what the title should be for this question. Please edit the title if there is a better way to describe this question. ============= Here's a definition from ...
0
votes
1answer
150 views

Convergence of $\sin(nx)$ in uniform and compact-open topologies

I came across this problem while preparing for a qualifying exam... not really sure where to start. Help is much appreciated :) Question: Consider the sequence $f_n(x) = \sin(nx)$ in ...
1
vote
1answer
80 views

Existence of a $\sup$ and local compactness in the countable ordinals $\Omega$

I understand how the set of countable ordinals $\Omega$ is not compact, but how is it locally compact? Also, how can the existence of a least upper bound ($\sup$) for a non-decreasing sequence ...
1
vote
2answers
422 views

Proof: in $\mathbb{R}$, $((0,1),|\cdot|)$ is not compact.

Let $(M,d)$ be a metric space, and $A\subset M$. By definition, $A$ is said to be compact if every open cover of $A$ contains a finite subcover. What is wrong with saying that, in $\mathbb{R}$, if ...
0
votes
1answer
110 views

Is the restriction of a continous function to a compact set uniformly continuous?

Suppose $f:S\to T$ which is continuous and consider $A\subset S$ which is compact, is $f$ is uniformly continuous on A? I have tried to prove it by considering a finite subcover of $A$ say $\bigcup ...
1
vote
1answer
36 views

Continuous function and the openess of points

Suppose that I have a continuous function $f: X \rightarrow Y$ such that $f(a) = f(b) $ where $a$ and $b$ are points of $X$. Is it the case that we have that either both $a$ and $b$ are open or ...
4
votes
2answers
130 views

Version of Jordan’s theorem for unbounded curves

Let $I$ denote the open interval $]0,1[$. Let $\gamma$ be a countinous map $I \to {\mathbb R}^2$. We say that $\gamma$ is stretched if it contains points that are arbitrarily close to the origin, ...
3
votes
1answer
222 views

Restriction of a continuous function to a compact subspace in one of its multiplicands.

I was practicing for a qualifying exam and came across this bugger. Let me know your thoughts. Question: Let $X, Y, Z$ be topological spaces, and let $f: X \times Y \to Z$ be a continuous function. ...
2
votes
1answer
435 views

non-vanishing k-form on a k-manifold in $\mathbb{R}^n$ implies orientability

I want to know how to prove the theorem: If M is a k-manifold in $\mathbb{R}^n$, then it is orientable if and only if there is a volume form defined globally on M. I'm currently stuck at this step: ...
2
votes
3answers
307 views

Cardinality of all dense and countable sets of $\mathbb{R}$

What is the cardinality of the following set: $$\mathbb{A}:=\{A \ : A\subseteq\mathbb{R} \ \ \text{dense and countable}\}$$ (Is $\mathbb{A}$ a separable space?) Thank You!
0
votes
1answer
139 views

Homeomorphism confusion

I stumbled upon this excerpt as I was reading Graph Theory by Reinhard Diestel: A polygon is a subset of $\mathbb{R}^2$ which is the union of finitely many straight line segments and is ...
8
votes
1answer
80 views

Is every good enough space homotopy equivalent to a compact one?

Well, the question is more or less completely contained in the title. I found a partially related question on MO, namely this, and googling around reveals an amazing theorem of Browder, Levine, and ...
5
votes
2answers
200 views

$f(\mathbb{R}\setminus \mathbb{Q}) \subseteq \mathbb{Q}$ and $f(\mathbb{Q}) \subseteq \mathbb{R}\setminus \mathbb{Q}$ imply that $f$ is not continuous [duplicate]

Possible Duplicate: No continuous function that switches $\mathbb{Q}$ and the irrationals Let $f: \mathbb{R} \to \mathbb{R}$ be function satisfying the two conditions: ...
1
vote
1answer
71 views

Is the Idele class group Hausdorff?

I was wondering if for a global field (function or number field) $K$, is $C_K$ Hausdorff? Thank you
0
votes
1answer
45 views

Interpolating arc in path-connected set of ${\mathbb R}^n$

I know that in ${\mathbb R}^n$ (as in any Hausdorff space), a path-connected subset $A$ is automatically arc-connected also. Is it also true that, given any finite subset $B$ of $A$, there is an arc ...
0
votes
2answers
139 views

Connectedness of set which is an intersection of some connected set [duplicate]

Is a set which is an intersection of some connected set still connected? I think it is not true but could not think of an example.
1
vote
1answer
147 views

Is $S^\circ$ convex if $S$ convex?

Suppose $S\subset \mathbb{R^n}$ and $S^\circ$ denoted as the interior of $S$.Is $S^\circ$ convex if $S$ convex? $S$ is Convex mean $ \forall x,y\in S, kx+(1-k)y\in S, k\in [0,1]$ I know how to prove ...
1
vote
2answers
50 views

What is the definition of limit of a sequence of sets?

Here's an example. Let $s\in \mathbb{Z}^+$ and $q=2s+1$. Define $T_{s,i}(x)=\frac{x}{q} + \frac{2i}{q}$ where $x\in\mathbb{R}$ and $i=0,1,\ldots,s$. Define recursively $I_{s,0} = [0,1]$ and ...
6
votes
2answers
162 views

Proof that $\omega^\omega$ is completely metrizable and second countable

I have almost solved the following problem but am stuck at the very end, can you help me finish it? Thank you for your help. Let $n<\omega$ and $t\in {}^n\omega$. We define $U_t=\{s\in ...
1
vote
2answers
43 views

Set $A$ and Set $B$ in each others closures

Let $A$ and $B$ be sets such that $A\subset \overline B$ and $B \subset \overline A$. Isn't that an equivalent statement to saying that $A$ is dense in the closure of $B$ and that $B$ is dense in ...
3
votes
1answer
524 views

A boundary point may not be an accumulation point proof

A boundary point $z∈A$ may not be an accumulation point. proof: If $z$ is an isolated point of $A$ (i.e. there is a ball $B(z,r)$ such that $B(z,r)\cap A=\{z\}$) then $z$ is a boundary point ...
1
vote
1answer
128 views

pre-Hilbert, Normed and Metric Spaces: a few questions about their definitions

A vector space $R$ , with fixed inner product $\langle x,y \rangle$ is called pre-Hilbert Space A vector space $E$ with fixed norm $||.||$ is called Normed Space. A set $X$ with fixed ...
4
votes
2answers
194 views

Is countable intersection of perfect set perfect? And one question about closed connected set

Let $\{A_n\}$ be a sequence of perfect sets in a metric space. Then, is $\bigcap_{n\in\omega} A_n$ perfect? Recently, i have studied some Cantor-like sets and i got this very natural question. Let ...
2
votes
1answer
67 views

Dense path in square, constructed from grid

Working on this stack overflow question led me to ask another, related question (here) . This first attempt was shown unsuccessful in the answer to it, so I try a rather different approach here. Let ...
1
vote
1answer
131 views

In regard to a retraction $r: \mathbb{R}^3 \rightarrow K$

Let $K$ be the "knotted" $x$-axis. I have been able to show that $K$ is a retract of $\mathbb{R}^3 $ using the fact that $K$ and the real line $\mathbb{R}$ are homeomorphic, $\mathbb{R}^3$ is a normal ...
17
votes
1answer
510 views

Given a group $ G $, how many topological/Lie group structures does $ G $ have?

Given any abstract group $ G $, how much is known about which types of topological/Lie group structures it might have? Any abstract group $ G $ will have the structure of a discrete topological group ...
1
vote
2answers
128 views

$\mathbb{R}P^3$ is homeomorphic to the lens space $L(2,1)$

Show that the $3$-dimensional real projective space $\mathbb{R}P^3$ is homeomorphic to the lens space $L(2,1)$. (I am not sure but the problem is probably from the book Knots and Links which is ...