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; ...

learn more… | top users | synonyms (2)

2
votes
1answer
40 views

Countable $T_2$-spaces that are invariant under point removal

Let's call a $T_2$-space $(X,\tau)$ invariant under point removal if for every $x\in X$ we have $X \cong (X\setminus\{x\})$ where $X\setminus\{x\}$ is endowed with the subspace topology. Examples of ...
1
vote
1answer
25 views

Initial topology

I have read that for all $U \in \sigma (X, {f_i, i \in I})$ (initial topology) there exists a finite number of open sets $V_i$ in $T_{Y_i}$ s.t. $U = \cap_{i\in G}f_i^{-1}(V_i)$ or $U = \cup_{i\in ...
4
votes
2answers
60 views

Prove that y>2x+1 is open?

The answer is inserted but what I'm looking for is a heavy breakdown on this. My professor tried to explain to me a harder version but I don't understand it. Solution: What my prof does for ...
3
votes
1answer
27 views

Show that the Sorgenfrey line does not have a countable basis.

I am trying to understand this proof from Munkres' book which shows that the Sorgenfrey line does not have a countable basis. His proof is: Let $\beta$ be a basis for $\mathbb{R}_l$. Choose for each ...
1
vote
1answer
43 views

This subcollection of the base is also a base

Let $B_n$ be a countable base of a locally compact Hausdorff space $X$. I am trying to show that $S = \{B_n \mid \overline{B_n} \text{ is compact} \}$ is also a base. I imagine the proof to be short ...
3
votes
2answers
165 views

Homeomorphic metric spaces

I want to examine if $(0,1] $ and $\mathbb R $ are homeomorphic. We work on metric space $(\mathbb R, e)$, where $e$ stands for the euclidean metric. My answer: Let's assume there is a ...
2
votes
0answers
38 views

Topology Bases and Real Numbers

Let $(X,\tau)$ be a topological space. Suppose that $\mathcal{C}$ is a subset of $\tau$, and for every $U$ in $\tau$ and every $x$ in $U$ there exists a $C$ in $\mathcal{C}$ such that $x \in C ...
0
votes
1answer
20 views

Homeomorphism of two sets

$X $ is a topological space with infinite cardinality which is homeomorphic to $X\times X$.Can we conclude that $X $ is homeomorphic to a subset of $\mathbb R$ ?
0
votes
0answers
18 views

Proving that $\left(X(v,v)\right)^{\frac{1}{2}}$ defines a norm on a vector space where X is a positive-definite bilinear form.

Want to show: If $X$ is a positive-definite bilinear form on a vector space $G$ with real-valued scalars and $v\in G$, then $\left(X(v,v)\right)^{\frac{1}{2}}$ defines a norm on $G$. Thus far I have ...
1
vote
2answers
36 views

If there exists a continuous non-constant map to the integers, then the space is not connected

Let X be a topological space. show that if there exists a continuous, non constant map from X to the integers with the discrete topology, then X is not connected. So I know that connected subspaces ...
0
votes
2answers
24 views

If ${p_{n}}$ is a sequence in a compact metric space $X$, then some subsequence of ${p_{n}}$ converges to a point of $X$.

This has been proved in Rudin, but I am trying a different approach. Please suggest me how to prove this with my approach. I have done till the following. Since $X$ is compact, every open cover of ...
0
votes
0answers
16 views

Quotiemt map about fundamental domain

Let X be a topological space. G is a group, with group action G$\times$X$\rightarrow$X. Let D be a closed subset of X s.t 1.$\cup_{g\in G}(g\cdot D)=X$ Here $g\cdot D$ is image of group element g ...
2
votes
1answer
21 views

Sequentially Compact of a closed ball in $l^p$

Consider a $l^p$-space and define a closed ball in $l^p$ as $$\bar B:=\left\{\{a_n\}_n: \sum_n^\infty |a_n|^p \le 1 \right\}$$ I was wondering why $\bar B$ is NOT sequentially compact? ...
1
vote
1answer
29 views

Whitney sum of smooth vector bundles

I was reading through Lee's smooth manifolds book, in his chapter on vector bundles. Upon reading about smooth vector bundles and its definition, I was wondering if the whitney sum of two smooth ...
1
vote
1answer
43 views

Topology Metrics

Let $X$ be a set, and suppose $d$ is a metric on $X$. Let $$\mathcal{B} = \{B(x,\epsilon) : x \in X,\space \epsilon > 0\}$$ be the set of open balls where $$B(x,\epsilon) = \{y \in X, \space d(x,y) ...
3
votes
2answers
67 views

Why the space S1 and S1/Z_2 is topologically identical?

I am a physicist studying liquid crystals. My research is bit related to topology but I don't have much knowledge of it. Recently I read from a the book Soft matter physics: An introduction that ...
3
votes
1answer
30 views

Defining a quotient topology based off of level sets.

Is my understanding of the quotient topology correct? Let $f$ be a continuous surjective mapping $f : \mathbb R^n \to \mathbb R$. Let $\bf x_1$ and $\bf x_2$ be $n$-tuples in $\mathbb R^n$. Say $\bf ...
0
votes
1answer
32 views

Normed space and convex hull of closed subset

Let $(V, ||\cdot||)$ be a normed space. If $ C\subseteq V$ is a closed set we do not know if $ch(C)$ is closed or not. The professor provided this example that as of now I'm not getting: Consider the ...
3
votes
1answer
59 views

A base of topology

Consider a space of smooth functions $C^{\infty}[a,b]$ and a set $$\tau=\left\{B(f,\varepsilon_0,\varepsilon_1...\varepsilon_r):f\in C^{\infty}[a,b],r\in\mathbb{N}\right\} $$ where $f$ is arbitrary ...
1
vote
1answer
55 views

Continuity of a function?

Let $f: (\mathbb N, e_\mathbb N) \to(\mathbb R,e)$, where $e$ stands for the euclidean metric and $$f(n)=\begin{cases} n,\, n\ge 2\\\\0,\, n=1\end{cases}$$ Is $f$ continuous? Firstly, I can ...
0
votes
0answers
17 views

Is the locale of filters on an arbitrary lattice compact?

A mathematician has claimed in a private email to me, that the lattice of filters on every lattice is compact. I have proved it only for distributive lattices. I need help for non-distributive case. ...
0
votes
2answers
29 views

Question about the closure of a set

How to find $\overline{A}$, where $A=\lbrace (-1)^n(1+\frac1n), n\in \mathbb{N}^*\rbrace$ Please give me the steps to do in order to find $\overline{A}$ Thank you
1
vote
1answer
25 views

Is this proof for a necessary condition for an element to belong to the boundary correct?

Hi I am new to constructing a mathematical proof in topology and want to know if this argument is sound. I am a self-learner and appreciate all help. I will just do the positive statement here. John ...
4
votes
2answers
57 views

A first-countable, compact space which is non-separable [duplicate]

Let $X$ be a space which is first countable and compact. Is $X$ necessarily separable? Is $X$ necessarily second countable?
5
votes
0answers
52 views

Proving a function is an open map, with limitations

Following from a previous question I posted in here, given the same thing, i.e. an open set $U$ in $\mathbb{R}^2$ and a continuous function $f:U\rightarrow \mathbb{R}^2$ such that for each $u\in U$ ...
6
votes
1answer
53 views

What's an easy way to show that $GL(n,\mathbb C)$ is connected? [duplicate]

I think I've to show it's path connected, but can't figure out the path functions explicitly. Can anyone give these path maps?
1
vote
1answer
31 views

Are the spaces of real orthogonal, complex unitary, hermitian or symmetric matrices connected?

I want to know which of these are connected and which are not. I think I've to take some continuous map from the set of matrices to $\mathbb R$ or $\mathbb C$ and interpret these matrix sets as ...
2
votes
1answer
25 views

Continuous Measures: Range

Let $\Omega$ be a sigma-finite measure space with no atoms. (Reminder: A subset $A\in\Sigma$ is an atom if $\mu(E)<\mu(A)$ implies $\mu(E)=0$ for all $E\subseteq A$.) Then the measure attains ...
2
votes
1answer
51 views

Topology from fundamental system of neighbourhoods once and for all

This problem is probably closely related to a bunch of similar questions touched upon elsewhere (see for example Going from a fundamental system of neighborhoods to a topology and vice versa and ...
1
vote
2answers
21 views

Closure of linear subspace in Topological vector space

Let $X$ be a TVS, $x\in X$ and $M<X$ be a linear subspace. Does $x\in M+U$ for every open neighborhood $U$ of $0$ imply that $x$ is in the closure of $M$? EDIT: This argument is used here: ...
6
votes
2answers
134 views

Meaning of a discrete topological sub-space?

Given a topological space $X$ and a set $U\subseteq X$, what is the meaning of $U$ being a discrete sub-space of $X$? I do know what a discrete space is, so as far as I understand it, the meaning is ...
2
votes
1answer
31 views

Topological Vector Space not induced by Metric

Can anyone give me an example of a Topological Vector Space that is not metrizable? I know that the neighborhood base of $0$ needs to be incountable, and all I can construct then is no topological ...
2
votes
0answers
36 views

Find $A^\circ , \, cl (A),\, A', \partial A$

Consider the set $Α=\left\{ \left(\dfrac 1 n, \dfrac 1m \right):\, n,m \in \mathbb N\right\}$. We want to find the sets $A^\circ=int\, A, \,cl(A) ,\, A' (\text {=derived set}) , \, \partial A $. I ...
1
vote
1answer
35 views

Let $A$ be a subset of metric space $X$ and $A^{'}$ be the set of limit points of $A$. Show that $A^{'}$ is closed.

I have done the following as a proof to this: Assume $A^{'}$ is open. $\forall x \in A^{'}$, $\quad \exists r>0$ such that $B_{r}(x)\subset A^{'}$. Also, since $x$ is a limit point of $A$, then ...
3
votes
2answers
39 views

Topological space which is not locally connected

In class we defined a locally connected space as a space that has a basis consisting of connected sets. I don't quite understand what a space which is not locally connected would look like. At least ...
0
votes
1answer
29 views

Is f continuous (i.e $f^{-1}(\,open)$ is open) $\iff$ $f(\,closed \,set \,in \,S)$ is a closed set in U?

Let $f:(S,T) \to (U,V)$, where $(S,T)$ and $(U,V)$ are topological spaces. Is f continuous (i.e $f^{-1}(\,open)$ is open) $\iff$ $f(\,closed \,set \,in \,S)$ is a closed set in U? If not is one ...
1
vote
3answers
41 views

Suppose $x$ and $y$ are two points in a metric space, such that $d(x, y) < 1/n$ for any natural $n$. Show that $x = y$

I have the following outline for my proof. Suppose $x$ is not equal to $y$. Then, $d(x,y)=a <1/n, $ $\forall n \in N$, where $a$ is a positive real. Now, $\ y \in B_{1/n_{1}}(x)$ for some $\ n_{1} ...
3
votes
4answers
44 views

Hausdorff Topological Space

Wanted to explain what I think a Hausdorff is in my own words because maybe that is the root of the problem. A Hausdorff Space is one in which for every x and y in X with x does not equal y, there ...
0
votes
1answer
36 views

A countably compact, locally connected space has finitely many components

The question is to prove that a countably compact locally connected space $X$ has finitely many components. My (incorrect) proof goes like this: Let $\{ B_i \}_{i = 1}^{\infty}$ be an open cover of ...
0
votes
0answers
23 views

Showing that the topologist's sine curve is not path connected using an argument of sequences.

I am familiar with many proofs of the fact that the set (defined below) is not path connected. My favorite uses the fact that $[0,1]$ is compact and another good one uses the intermediate value ...
2
votes
2answers
38 views

Can there be a homeomorphism of $[0,1]$ that maps $\mathbb Q$ to the dyadic rationals?

Can there be a homeomorphism of $[0,1]$ to itself that maps $\mathbb Q$ to the dyadic rationals? What I know is that homeomorphisms of $[0,1]$ have to be increasing or decreasing and map endpoints to ...
2
votes
3answers
33 views

Proof with intersection of closed A set and interior of B set

I'm stuck at this proof. Let $\mathrm{A}$ and $ \mathrm{B}$ be disjoint sets in topological space $ (\mathrm{X}, \tau) $. Prove that if $\mathrm{A}$ is open set in $\mathrm{X}$ then $ ...
3
votes
2answers
30 views

Continuity of a function through adherence of subsets

We say two sets $A,B$ being $adherents$ if we have $(\overline{A} \cap B)\cup(A\cap\overline{B})\neq \emptyset $. Prove that a function $f:X\to Y$, with $X,Y$ topological spaces, is continuous if ...
0
votes
1answer
44 views

continuous function from one metric space to another metric space

Is differentiation $f(x) \rightarrow f'(x)$ a continuous function from $C^1[a,b] \rightarrow C[a,b]$ ? Is integration $f(x) \rightarrow \int_a^x \! f(t) \, \mathrm{d}t $ a continuous function from ...
0
votes
1answer
48 views

Topology. Understanding what a base is intuitively

A collection $\{V_n\}$ is said to be base for $X$ if the following is true: For every $x$ that's an element of $X$ and every open set $G$ that is a subset of $X$ such that $x$ is an element of $G$, we ...
1
vote
1answer
73 views

Proof of theorem 26.15 in “General Topology” by Willard

26.14 Definition. A simple chain connecting two points $a$ and $b$ of a space $X$ is a sequence $U_1,\ldots,U_n$ of open sets of $X$ such that $a\in U_1$ only, $b\in U_n$ only, and $U_i\cap ...
3
votes
1answer
44 views

Different identifications of the same sides of a polygon make the same quotient space

Let P$\subset\mathbb{R}^{2}$ be a polygon with sides $l_{1},...,l_{n}$ parametrized by the curves $\alpha_{1}(t),...,\alpha_{n}(t)$. Let $\beta_{1}(t),...,\beta_{n}(t)$ be another parametrization of ...
1
vote
0answers
32 views

Fix point theorem for measures? metric on space of measures?

I have the following problem: I consider a probability space $(\Omega, \mathcal{F}, \mu)$ where $\Omega= C_0([0,1])$ (space of continuous functions on $[0,1]$ starting from 0), $\mathcal{F}$ is a ...
1
vote
2answers
27 views

How do open subsets of X/G look like?

Let $G$ act continuously on $X$, where $X$ is a topological space. So I wonder about how open subsets look like in $X/G$. The action $a$ is defined as $a(g,x)=g.x$.
2
votes
1answer
26 views

question on existence of open set

Let $U$ be a bounded open set in $\mathbb{R}^n$ and $A$ be an open subset of $U$. Fixed $\epsilon >0$. Does there exist an open set $B \subset U$ such that $B \cap \overline{U} \ne \emptyset$ and ...