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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7
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4answers
408 views

A kind of converse to the Jordan curve theorem

The Jordan curve theorem in $\mathbb{R}^2$ says that if $S$ is a closed curve in $\mathbb{R}^2$. Then $S$ splits $\mathbb{R}^2$ into exactly two connected components $A$ and $B$. I was thinking about ...
1
vote
1answer
20 views

on the hyperspace K(X) with Vietoris topology

Let $X$ be a Polish Space without isolated points. Let $K(X)$ be the sets of the compact non empty subsets of $X$ equipped with the Vietoris topology. Can we say that also $K(X)$ has no isolated ...
1
vote
1answer
33 views

Open sets in the unitary group $ U(\mathcal{H}) $ of a Hilbert space $ \mathcal{H} $.

Let $H$ be an infinite dimensional Hilbert space and let $(x_i)_1^\infty$ be an orthonormal basis for $H$. Consider $U(H)$ the unitary group of the continuous unitary operators on $H$. Equip $U(H)$ ...
0
votes
0answers
43 views

Compact 1-manifolds

I want to find all compact 1-manifolds. I think these will be lines, with fixed endpoints, and also the case where the endpoints meet. Hence I think all compact 1-manifolds are homeomorphic to the ...
1
vote
1answer
46 views

uncountable co-meagre set in Polish Spaces

Let $X$ be an uncountable Polish Space and let $Y$ be a co-meagre subset of $X$. How can I prove that $Y$ is uncountable? Possibly proof without using borel sets. Thank you
0
votes
4answers
25 views

Proving path-connectedness of $\mathbb{R}^2\setminus\mathbb{E}$ where $\mathbb{E}$ is the set of points with both coordinates rational

Let $\mathbb{E}$ be the set of all points in $\mathbb{R}^2$ having both coordinates rational. Prove that the space $\mathbb{R}^2\setminus\mathbb{E}$ is path-connected. I think I just need to ...
0
votes
3answers
54 views

Topology on vector space such that vector operations are not continuous

everyone I am reading about distribution theory and the first concept is topological vector space. The definition says it is a topological vector space if its topology keeps the two vector ...
2
votes
2answers
57 views

Prove that $f:\mathbb{R}^n\to\mathbb{R}$ is continuous

Let $f:\mathbb{R}^n\to\mathbb{R}$ such that for every continuous curve, $\gamma:[0,1]\to\mathbb{R}^n$: $f\circ\gamma$ is continuous. Prove that $f$ is continuous. So I know we shall prove it by a ...
2
votes
1answer
28 views

Making a Klein bottle from 2 Möbius bands

I thimk this can be done by idemtifying points on the boundary but I am not sure how to show this Any ideas? E.g. By drawing nets..
0
votes
1answer
40 views

A equal closure of B

Let $A,B\subset X$ be subspaces of a normed space. I have quite a general question: If I want to show that $A=\bar{B}$ where $\bar{B}$ is the closure of $B$, is it enough to show that first ...
0
votes
3answers
36 views

Show that $X $ is a infinite set ,it is connected in the finite complement topology?

what i had done :let X be a infinite set and there be open sets $A,B$ on it. such that $A\cap B =\varnothing$ or $A\cap B\ne\varnothing$then $X-A\cap X-B \ne \varnothing $ .which means there doesn't ...
0
votes
1answer
41 views

Graphe of a continuous function

How to prove that if F is Hausdorff and $f: E\rightarrow F$ is continuous the complement of $G(f)$ is open, $G(f)=\{(x,f(x)), x\in E\}$ Without using the diagonal $\Delta$ i want to deduce that ...
6
votes
2answers
52 views

When An Infinite Product Topology Is Hausdorff?

Is the following true: Suppose $(X_1, \tau_1)$ is Hausdorff while $(X_2, \tau_2)\space ...$ are not. $x, y \in X_1 \space\space U,V\in\tau_1\space x \in U , y \in V \space\space V \cap U = \emptyset ...
3
votes
0answers
34 views

$\mathbb{R}^n$ bundle over $B$ being compact and having finite covering dimension implies has finite type? [closed]

Let $\xi$ be a $\mathbb{R}^n$-bundle over $B$. If $B$ is paracompact and has finite covering dimension, does it follow that every $\xi$ over $B$ has finite type?
2
votes
1answer
55 views

Prove that the following statements are equivalent

Let $f: (X,d) \rightarrow (Y, d')$ be a function. Prove that the following are equivalent: $f$ is continuous . For every $A \subset X$, $f(cl(A)) \subset cl(f(A))$. For every closed set $B$ in ...
1
vote
2answers
44 views

Can someone help me understand proof that “a sphere is connected”

The question is to prove Prove $S = \{(x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2 = 1\}$ is connected. The answer provided as: The sphere can be represented as a union of meridians (each of ...
1
vote
1answer
32 views

No subsequence converges when sequence has limit point in topological space

In general topological space, suppose $x_0$ is a limit point of a sequence $\{x_n\}$, is it possible that there is no subsequence that converges to it?
1
vote
1answer
39 views

What symbol is used for product topology?

Let $((X_k,\tau_k))_{k \in N}$ be topological spaces. The product topology $\tau$ on $X = \prod_{k \in N} X_k$ is the coarsest topology that makes all projections $\pi_k:X \to X_k$ continuous. Is ...
0
votes
0answers
22 views

topological property of a completely metrizable space

Let $X$ be a completely metrizable space. Let $K$ be a compact of $X$ and let $C$ be a closed of $X$. Suppose there is an open set of $X$, say $V$, such that $K\cap C\subseteq V$. Then there is an ...
2
votes
2answers
83 views

Are $\mathbb{C}^2$ and $\mathbb{C}^2/(x,y)\sim(y,x)$ homeomorphic?

Let $A$ be the set of monic quadratics over $\mathbb C$ and let $B$ be the set of unordered pairs over $\mathbb C$ where possibly the two elements of the pair may be the same. Then the map which takes ...
1
vote
0answers
26 views

Uniform continuity with respect to parameter.

Let $\mathbb{X},\mathbb{Y}$ and $T$ metric spaces. A family $\{f_t\}_{t\in T}$ of maps $f_t:\mathbb{X}\to\mathbb{Y}$ is uniformly continuous with respect to parameter $t$ if, $$ (\forall ...
0
votes
1answer
16 views

Show that every norm is a $1$-lipschitz function

Let $\|\cdot\|_0$, a norm on $\mathbb{R}^n$. Show that the function $\|\cdot\|_0$ is $1$-lipschitz and hence, continuous. Meaning, I need to prove that: $$\big|\|x\|_0-\|y\|_0\big| \le \|x-y\|$$ ...
2
votes
1answer
17 views

compactness requirement for the tube lemma of a product space.

The tube lemma: Let $X,Y$ be topological spaces s.t $Y$ is compact. Let $X_0 \in X$ and let $N$ be an open set in $X \times Y$ so that $x_0 \times Y$ is contained in $N$. Then there exits a ...
0
votes
1answer
58 views

Exercise 6 in section 50 in Topology textbook of Munkres.

I am looking for a reference of the proof of the following thoerem in Munkres it's exericse 6 in section 50 page 315 and it goes as follows: Let $X$ be a locally compact Hausdorff space with a ...
1
vote
0answers
47 views

Show that in a topological space $\partial\partial A \subset \partial A$ [duplicate]

I have the following task: Show that in topological space $\partial\partial A \subset \partial A$, where $\partial A$ denotes the set of boundary points of $A$, that is $\partial A = ...
0
votes
0answers
22 views

Is $B_1 \times B_2 \times … \times B_n \times X \times X \times …$ a base for the product topology?

As the title asks, Is $B_1 \times B_2 \times ... \times B_n \times X \times X \times ...$ a base for the product topology? (for all $B_n \in \tau$) i.e. Can you assume that the open sets $B_n$ ...
3
votes
1answer
50 views

Structure of the $L_1$ space of measurable subsets of $[0,1]$

Let $\mathcal A$ be a Borel $\sigma$-algebra on $[0, 1]$, and let's introduce a metric on it by $$ d(A, B) = \lambda(A\mathbin\Delta B) \qquad \forall A,B\in \mathcal A $$ where $\lambda$ is the ...
0
votes
1answer
55 views

Use Existence Theorem to determine if $x = \frac{1}{4-t^2}$ yields a solution/s to $\frac{dx}{dt} = 2tx^2$. If so, give them. [duplicate]

I believe this is not a duplicate because the $D$ is different. Definition: Let $x = x(t)$. A solution of the first order ODE $x' = f(t,x)$, where $f$ is defined on some domain $D \subseteq ...
0
votes
0answers
25 views

Infinite intersection of frontiers

Let $(X,d)$ be a compact metric space and $f\colon X\to X$ a homeomorphism. Let $\delta >0$, define closed sets $B_n=D[f^n(x),\delta]$ (closed ball of center $f^{n}(x)$ with radius $\delta$ in ...
1
vote
1answer
31 views

Closed set, open set or neither?

Just a quick question - is a straight line that goes on indefinitely viewed as a closed set, open set or neither? Seeing as it includes all the boundary points as it travels, but it doesn't have any ...
3
votes
1answer
43 views

Product of compacts is compacts using closedness of projection to second component?

A well known characterization of compactness says $X$ is compact iff for all spaces $Y$ the projection $X\times Y\rightarrow Y$ is a closed map. I'm wondering whether there's some simple formal way ...
0
votes
2answers
19 views

Connected/No connected set in $\mathbb{R}$, application of Munkres definition.

in $\mathbb{R}$, with the usual topology, is the set $A = \mathbb{[a,b]}$ connected? what about $B = [a,b] \cup [c,d]$ where $a < b < c < d$, i would say $A$ is connected while $B$ is not. ...
0
votes
0answers
28 views

Why do these IVPs have a unique maximal solution? What is the largest possible domain for each ODE?

Definition: Let $x = x(t)$. A solution of the first order ODE $x' = f(t,x)$, where $f$ is defined on some domain $D \subseteq \mathbb R^2$ s.t. $D$ is open and connected, is a differentiable ...
2
votes
0answers
29 views

Why is a continuous, degree zero and odd function from $S^n$ an antipodal map?

https://terrytao.wordpress.com/2008/11/27/the-kakeya-conjecture-and-the-ham-sandwich-theorem/ In this blog post, Terry tao states that "It is easy to see that $f$ is continuous, homogeneous of degree ...
-4
votes
0answers
69 views

Use Existence Theorem to determine if $x - \ln(x) = t^2 + 1$ yields a solution/s to $\frac{dx}{dt} = \frac{2tx}{x-1}$. [duplicate]

I believe this is not a duplicate because the $D$ is different. Definition: Let $x = x(t)$. A solution of the first order ODE $x' = f(t,x)$, where $f$ is defined on some domain $D \subseteq ...
-2
votes
0answers
48 views

Use Existence Theorem to determine if $t^2 - \sin(t+x) = 1$ yields a solution/s to $\frac{dx}{dt} = 2t\sec(t+x)-1$. [duplicate]

I believe this is not a duplicate because the $D$ is different. Definition: Let $x = x(t)$. A solution of the first order ODE $x' = f(t,x)$, where $f$ is defined on some domain $D \subseteq ...
1
vote
1answer
20 views

Quotient metric space

Let $X$ be some set, $(Y, \rho)$ be a metric space and $f:X\to Y$ be some map. Let $d$ be a pseudometric on $X$ defined by $d(x', x'') = \rho(f(x'), f(x''))$ and consider a quotient metric space ...
1
vote
2answers
95 views

Use Existence Theorem to determine if $e^{tx} + x = t - 1$ yields a solution/s to $\frac{dx}{dt} = \frac{e^{-tx} - x}{e^{-tx} + t}$.

Definition: Let $x = x(t)$. A solution of the first order ODE $x' = f(t,x)$, where $f$ is defined on some domain $D \subseteq \mathbb R^2$ s.t. $D$ is open and connected, is a differentiable ...
3
votes
2answers
33 views

Show that $F\subset Y$ is closed in $Y$ iff $F=Y\;\cap\;H$ where $H\subset X$ is closed in $X$.

I have the following problem: Let $(X,\tau)$ be a topological space and $(Y,\tau_y)$ be its subspace, that is $Y\subset X$ and $\tau_y=\{\;A\;\cap\;Y\mid A\in\tau\;\}$. Let also $F\subset Y$ ...
0
votes
0answers
31 views

inverse image of a function in topology

I really got confused when I study continuous mapping in topology. one of proof uses following statement. Can you please help me understand this. Let $(X,\tau)$ and $(X,\tau_1)$ be topological spaces ...
1
vote
1answer
118 views

Prob. 21, Sec. 17 in Munkres' TOPOLOGY, 2nd ed: Closure and complementation can result in at most 14 different sets

Here's Prob. 21 in Sec. 17 of the book Topology by James R. Munkres, 2nd edition: Consider the collection of all subsets $A$ of the topological space $X$. The operations of closure $A \to ...
1
vote
0answers
33 views

Is uniform continuity a property of the category of completely regular spaces?

If $(X, U)$ and $(Y, V)$ are uniform spaces then one has the notion of a map $f : X \to Y$ to be uniformly continuous relative to $U$ and $V$. A uniform space $(X, U)$ induces a completely regular ...
0
votes
1answer
41 views

A counter example for the homeomorphism between quotient product of coproduct and the space itself

I need an example that: For sets $X,Y$ in $\mathbb{R}$, s.t $X\cup Y=\mathbb{R}$, and $X\sqcup Y/\sim$ is not homeomorphic to $\mathbb{R}$, where $\sim$ means identifying the $x\in X$ and $y\in Y$ if ...
5
votes
2answers
144 views

How to picture a first countable space?

I find myself forgetting what it means for a space to be first countable on a frequent basis. This is unlike say other terminologies such as "Hausdorff space", where you can picture balls separating ...
0
votes
0answers
16 views

In $\mathbb{R}_{\ell}$ and finite complement topology, to what point(s), if any, does the sequence $x_{n} = 3 + \frac{(-1)^{n}}{n}$ converge?

In the finite complement topology on $\mathbb{R}$, to what point or points (if any) does the sequence $x_{n} = 3 + \frac{(-1)^{n}}{n}$ converge? What about in the lower limit topology ...
-2
votes
1answer
51 views

’Stereographic projection’

This is problem 1.12 in Armstrong's Basic Topology: ’Stereographic projection’ $\pi$ from the sphere minus the north pole to the plane. Work out a formula for $\pi$ and check that $\pi$ is a ...
0
votes
0answers
35 views

Cluster point vs limit point in net

In topological space $X$, the cluster point $s$ of a net $S$ is that $S$ is frequently in every neighborhood of $s$. It is obvious that if $s$ is a cluster point of $S$, then it is the limit point of ...
1
vote
1answer
40 views

Proof for sets and functions.

I have been proving problems like this all day with ease, but this is is just puzzling to me. Where do I start? Also, a site with questions and answers to problems like these.
0
votes
1answer
47 views

Why does this equality imply that the inverse image is open?

Question: If $A$ is a closed subset of a metric space $X$, show that any map $f : A → \mathbb{E}^n$ can be extended over X. Solution: Let $p_i$ be the projection onto the i-th coordinate. Each ...
0
votes
1answer
34 views

$(0,1]$ is connected in relative topology. Different proof

The interval $X=(0,1]$ is connected in $\mathbb{R}$ w.r.t the relative topology. I am trying to show that $\emptyset, X$ are the only subsets which are both open and closed (I have seen the direct ...