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)

10
votes
5answers
7k views

Compact sets are closed?

I feel really ignorant in asking this question but I am really just don't understand how a compact set can be considered closed? I mean by definition of a compact set it means that given an open cover ...
2
votes
3answers
141 views

Finding the rank of subgroups of free groups?

How do you find the rank of a subgroup(of finite index) of a free group? I was thinking of looking at the fundamental group of a graph.
1
vote
1answer
106 views

Tychonov's theorem in countable spaces!?

Give a direct proof of Tychonov's theorem: If $(X_n, d_n)$ is compact, then $\left(\prod_{n\geq1} X_n, d\right)$ is compact.
1
vote
1answer
239 views

Complete metric space - sequences

The problem I'm trying is to prove whether or not the metric space of real-valued sequences $(x_n)$ such that $x_n=0$ for all but finitely many values of $n$, with the sup metric: $d((x_n),(y_n)) = ...
2
votes
2answers
493 views

Prove that the unit open ball in $\mathbb{R}^2$ cannot be expressed as a countable disjoint union of open rectangles.

Prove that the unit open ball in $\mathbb{R}^2$ cannot be expressed as a countable disjoint union of open rectangles. Open rectangles in $\mathbb{R}^2$ are subsets of the form $(a,b)\times(c,d)$. ...
1
vote
1answer
122 views

Why is sequence important for general topology

I wanna ask why sequence is important in general topology. As far as i know, many theorem can be proved without using sequence. Does sequence make some proof easier than other way? or is there any ...
1
vote
1answer
111 views

Stone-Cech compactification $\beta\mathbb N$ and convergence

In $\beta\mathbb N$ there are no non-trivial convergent sequences. I want to show this, but what is the meaning of non -trivial convergent sequence?
0
votes
0answers
69 views

Surface of a 2-sphere expressed as union of two closed disks

I'm reading a First Course in Differential Geometry by Chuan-Chih Hsiung and on page 8 he says "A closed disk that is homeomorphic to $I^2$ [i.e. $I\times I$, where $I = [a, b]$] is connected. The ...
1
vote
1answer
95 views

These quotient spaces are homeomorphic?

If $h:A\to B$ is a homeomorphism, where the subset $A_1$ of $A$ is homeomorphic to a subset $B_1$ of $B$. How can I prove that the quotient spaces $A/A_1$ and $B/B_1$ are homeomorphic? Thanks
13
votes
3answers
1k views

Is every connected metric space with at least two points uncountable? [duplicate]

As the topic, how to prove that every connected metric space with at least two points uncountable? Of course i know the definition that a countable set mean there is a bijection between the set and ...
1
vote
1answer
115 views

Gillman-Jerison Theorem

How can i prove it? [Gillman and Jerison] If a dense subspace $Y$ of a Tychonoff space $X$ is $C-embedded$ in X, then $Y$ is $ G‎‎_{\delta‎‎‎}-dense‎ $‎ in $X$.
0
votes
0answers
138 views

Are all even group actions actions on top spaces homeomorphisms?

I'd like to know if an even group action on a topological space is necessarily a homeomorphism. In particular, we say an action $G \times X \to X$ is even if, for any $x \in X$, there is an open ...
3
votes
1answer
164 views

Filters on topology

Show that the filter $\mathscr F$ has $x$ as a cluster point iff $x \in\bigcap_{F \in \mathscr F } \overline F$. For the the Proof of the 1st direction $(\Rightarrow)$ : Let the filter $\mathscr F$ ...
1
vote
1answer
85 views

Equivalent definition of a closed set.

I try to prove the equivalence between "C is closed in an ordinal $\alpha$" and "each strictly increasing sequence of elements of C of length $< cf\alpha$ converge in C". For $\Rightarrow$, it's ...
0
votes
2answers
366 views

Closed surjective map

How to prove that every closed surjective map is open? (Exercise from book Borisovich "General topology") Thank you very much!
4
votes
2answers
69 views

Characterisation of Cantor-connectedness

For Cantor-connectedness I use the following definition: A $p$-metric space $(X,d)$ is Cantor-connected if for any $\epsilon > 0$, any two points $x, y \in X$ can be connected by an ...
2
votes
1answer
73 views

Classification of the compactifications of $\mathbb{R}$

Do you know if there exists a classification of the compactifications of $\mathbb{R}$? From here, we can deduce that there exist only two compactifications with finite remainder: $[0,1]$ and ...
1
vote
1answer
102 views

$\sigma$-algebra on the space of all probabilty measures of a measurable space

I am trying to understand the arguments in a book I am reading. Consider the probability space $\left( X, \mathcal{B} \right)$ and let $\mathcal{P}$ be the set of probability measures on it. Let ...
4
votes
0answers
67 views

Inside out and backwards - more than a question of underwear…

While I am not a mathematician, I find a fascination for motion and 3D space. Lately this idea keeps turning over. Each time I put my underwear on and find the back label correctly in back but find ...
1
vote
1answer
590 views

Intervals are connected and the only connected sets in $\mathbb{R}$

As the topic, prove that Intervals are connected and only connected in $\mathbb{R}$. I know what is the definition of connected set. But not sure how to prove that.
11
votes
2answers
155 views

pseudo-inverse to the operation of turning a metric space into a topological space

The construction of turning a metric space $(X,d)$ into a topological space by inducing the topology generated by the open balls gives rise to a functor $Met\to Top$ for any reasonable category $Met$ ...
0
votes
1answer
79 views

Existence of the empty set in the standard d-metric topology on a set $X$.

Let $d$ be a metric on a set $X$, and let $$ B=\{B(p,e) = \{y\in X \mid d(p, y)<\epsilon \}\text{for every $p\in X$ and every $\epsilon>0$}\} $$ For $B$ to be the basis of a topology on $X$, ...
0
votes
1answer
32 views

Showing that $\mathbb{R},\mathcal{T} = \{]-a,a[ \mid a > 0\} \cup \{\emptyset,\mathbb{R}\}$ is a topology

I have to show that $\mathbb{R},\mathcal{T}$ is a topology, where $\mathcal{T} = \{]-a,a[ \mid a > 0\} \cup \{\emptyset,\mathbb{R}\}$. However, $$\cup_{n \in \mathbb{N}} ]-2 + \frac{1}{n} , 2 ...
2
votes
1answer
204 views

spaces for which every open set is a union of closed sets

In the context of generalized metrizability of spaces I'm interested in the property of a topological space that every open set in it is a union of closed sets. A google search led me to: ...
0
votes
1answer
94 views

non-Compact metrizable implies a countable closed discrete subspace?

I'm working through a proof in which $(X,\tau)$ is a $T_1$-space that is metrizable. The author says, "since $(X,\tau)$ is metrizable, there is a countably infinite, closed (in $X$) and discrete ...
2
votes
1answer
144 views

homeomorphism between the usual hawaiian ring and the onesided hawaiian ring

I am trying to come up with one. I can see that if you reflect the two sided rings and project them onto any one side, you get the one sided ring. However, I cannot think of a function that does not ...
1
vote
1answer
128 views

Topological definition of continuity on a real valued function with removable discontinuity

Consider some constant function $f(x)=c$, $f(x_0)=0$, for $c\neq 0$. This function is obviously discontinuous as $x_0$, so according to the topological definition of continuity, there must exist an ...
0
votes
1answer
216 views

Higher homotopy groups!

How would you show that $\pi_n, n>1$ of the Klein bottle is the trivial group? I was thinking Seifert-Van Kampen could be applicable?
6
votes
1answer
2k views

topology puzzle - without cut the rope, separate two rings

hello I wonder whether this puzzle is possible to solve. if possible, what kind of thing should I learn to solve this? the problem is make left one to right one without cut the rope only stretch and ...
1
vote
2answers
227 views

$Conv(Ex((C(X))_1))$ is dense in $(C(X))_1$?

Let $X$ be compact Hausdorff, and $C(X)$ the space of continuous functions over $X$. Denote the closed unit ball in $C(X)$ by $(C(X))_1$, then it can be shown $f$ is an extreme point of $(C(X))_1$ if ...
0
votes
1answer
51 views

differential forms proving degree can't be greater than dimention and 1-forms are linearly independent.

Every differential form w, say of degree k on an open set in R^n, can be written uniquely as $w = \sum_I a_I (x)dx^I$ , where the sum is over all possible lists $I$ of $k$ increasing indices, say $I ...
0
votes
1answer
121 views

Implicit function theorem => continuously differentiable functions

Consider the curve in $R^3$ consisting of the intersection of the paraboloid $z=x^2 + y^2$ and the cylinder $x^2 + y^2 = 1$. Near which points of this curve does the implicit function theorem say we ...
4
votes
0answers
60 views

topology on a graphs space

Let $\mathcal{G}$ be the set of locally finite, connected rooted graphs $(G,v)$ up to isomorphism $\cong$. Denote by $[G,v]_r$ the sub-graph of $(G,v)$ induced by the vertices at distance $\leq r$ ...
2
votes
1answer
111 views

Connected components of the complement of a connected component

Sorry for the "crazy" title, but that's what it is... Let $X$ be a topological space. Let $E$ be a subset of $X$. Let $F$ be a connected component of $E$. Is the following true? Why? For every ...
0
votes
0answers
149 views

Is the Hawaiian Earring connected? If yes, is it path connected?

Really stumped on this problem. How should I go about showing that the earrings are connected? Just claim that a circle is connected? To show it is path connected, we need to cook up a function that ...
3
votes
3answers
1k views

Closure of continuous image of closure

Let $f:X \to Y$ be a continuous map between topological spaces and $A \subset X$. Is it true that $\overline {f( \overline A)}= \overline {f(A)}$?
3
votes
3answers
121 views

Does $C_0(X)$ completely determine $X$?

Let $X$ and $Y$ be compact metric spaces. Let $C_0(X)$ and $C_0(Y)$ be the Banach spaces of continuous real-valued functions over $X$ and $Y$, respectively. If $F : X \rightarrow Y$ is a ...
1
vote
1answer
218 views

Prove that $[0,\infty)$ is not a manifold.

Prove that $[0,\infty)$ is not a manifold. Using diffeomorphisms and the implicit function theorem perhaps.
0
votes
3answers
380 views

Show a subset of $\mathbb{R}^2$ is connected.

Question: Show that the set $C=\{(x,y)\in \mathbb{R}^2: 1\leq x^2+y^2<2\}$ is connected. My Question: My main question is what open sets we should pick in $\mathbb{R}^2$. Once I know what open ...
1
vote
1answer
132 views

Prove that a sequence $\{x_k\}_{k=1}^\infty\subset \mathbb{R}^n$ converges to $x$ if and only if the map $ f(j) = x_j$ is continuous.

Need to know how to prove that a sequence $\{x_k\}_{k=1}^\infty\subset \mathbb{R}^n$ converges to $x$ if and only if the map $ f:\{1,2,3...\} \to\mathbb{R}^n$, $ f(j) = x_j$, is continuous. It's been ...
0
votes
0answers
102 views

How do I prove this surface is homeomorphic to the sphere

Let $S$ a compact and connected surface. If $S=U_1\cup U_2$, where $U_1,U_2$ are of finite character and the boundary of $U_1$ is in $U_2$. How can I prove that S is homeomorphic to the sphere? I'm ...
0
votes
1answer
100 views

Transferability of space properties via continuous functions

Let $f:(X, \tau_x) \to (Y,\tau_y)$ be a continuous and onto function. I need to show that if $X$ is separable (Lindelöf), then $Y$ is respectively separable (Lindelöf).
0
votes
1answer
98 views

point set topology-Metric spaces

Consider the two point set $X=\{a,b\}$ The possible topologies that can be found from X as follows. $$\begin{eqnarray} \tau_1&=&\{X,\emptyset\} &\text{Indiscrete topology} \\ ...
5
votes
2answers
2k views

Is the set of polynomials of degree less than or equal to $n$ closed?

This question is in relation to the space $C(I)$, $I = [a, b]$. Define $P_n =\{ a_0+\dots+a_nx^n \mid a_i \in \mathbb{R}\}$ (any or all $a_i$ could be zero); clearly $P_n \subset C(I)$. The norm I'm ...
3
votes
1answer
202 views

Covers, and compact sets

Please go to this lecture(1). I have numbered my questions for organization and added the picture which I will constantly refer to. I also apologize in advance for this lengthy question Right about ...
2
votes
1answer
180 views

Quotient space homeomorphic to $\mathbb{S^{1}} \times \mathbb{S^{1}}$

I am trying to solve the following topology problem: In $\mathbb{R^2}$ consider the unitary square $\mathbb{Q}:=[0,1]^{2}$. Defined is an equivalence relation by: $(x,y)\sim (x', y')\Leftrightarrow ...
4
votes
3answers
465 views

Locally Compact Hausdorff Space That is Not Normal

Someone told me that locally compact Hausdorff spaces (unlike compact ones) need not be normal. Can one give me please such an example? Thank you.
2
votes
3answers
547 views

If a set contains all its limit points must it be closed?

If a set $X$ in a topological space $T$ has the property that for all sequences $x_n \in X, x_n \to x \implies x\in X$ must X be closed? I know this is true for metric spaces but is it true for a ...
1
vote
1answer
267 views

Homeomorphism of product of topological spaces

I am stuck on a problem about homeomorphic topological spaces and can't go on... So the problem is: If we have that $X_{1} \times X_{2}\simeq Y_{1} \times Y_{2}$ (the product of topological spaces X1 ...
0
votes
1answer
97 views

Decreasing sequence of closed set in a metric space is convergent?

Let $\{E_n\}$ be a collection of bounded and closed subsets in a metric space $X$ such that $E_{n+1} \subset E_n$ and $lim_{n\to\infty} diam E_n = 0$. It's a theorem that if $X$ is complete, then ...