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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1answer
96 views

Is Steinhaus theorem ever used in topological groups?

Steinhaus theorem in $\mathbb{R}^d$ says that for $E\subset\mathbb{R}^d$ with positive measure, $E-E:=\{x-y:x,y\in E\}$ contains an open neighborhood of the origin. And for locally compact Hausdorff ...
7
votes
6answers
1k views

Topology: Example of a compact set but its closure not compact

Can anyone gives me an example of a compact subset such that its closure is not compact please? Thank you.
2
votes
1answer
408 views

Good Pairs in Algebraic Topology

Hatcher’s book says that $ \left( \mathbb{D}^{n},\mathbb{S}^{n - 1} \right) $ is a good pair; that is, there exists an open neighborhood $ V $ of $ \mathbb{D}^{n} $ containing $ \mathbb{S}^{n - 1} $ ...
2
votes
2answers
226 views

Accumulation points of the set $S=\{(\frac {1} {n}, \frac {1} {m}) \space m, n \in \mathbb N\}$

The exercise is to find the accumulation points of the set $S=\{(\frac {1} {n}, \frac {1} {m}) \space m, n \in \mathbb N\}$ I'm trying to prove that if $A$={accumulation points of the set $S$}, then ...
5
votes
2answers
906 views

Simply connected does not imply contractible. Is there a nice counter example in $R^2$?

The standard counter example to the claim that a simply connected space might be contractible is a sphere $S^n$, with $n > 1$, which is simply connected but not contractible. Suppose that I were ...
5
votes
3answers
164 views

A quotient map from $[0,1]$ to $S^1$

I would like to show that the function $f(x) = (\textrm{cos}2 \pi x, \textrm{sin}2 \pi x)$ is a quotient map; I have already shown that it is surjective and continuous (the latter by invoking the ...
4
votes
1answer
56 views

A question on spaces with calibre-$\aleph_1$

Suppose that $X$ is the $T_1$ space with $k$-in-countable base and $\aleph_1$ is a caliber of $X$. Must $X$ be second countable? Thanks for any help. A topological space has calibre $\aleph_1$ if ...
4
votes
2answers
149 views

Could someone explain me the significance of Brouwer's Fixed Point Theorem?

Well, I've been reading the proof of Sperner's Lemma and its use in proving Brouwer's Fixed Point Theorem (including the Coffee stiffing analogy ;-)). But I fail to understand the significance of this ...
1
vote
2answers
57 views

If $(X, \mathcal{D})$ is a discrete space and $(Y, \mathcal{T})$ is any topological space than any $f:X \rightarrow Y$ is continuous

The problem defines $f:(X,\mathcal{D}) \rightarrow (Y,\mathcal{T})$ where $\mathcal{D}$ is a discrete space and $\mathcal{T}$ is any topological space. I have to show that f is continuous. What I ...
5
votes
1answer
33 views

$X=\{a,b,c\}$ and $\mathcal{T}=\{X, \emptyset, \{a\}, \{b\}, \{a,b\}\}$. Determine if $f:X \rightarrow X$ is $\mathcal{T}-\mathcal{T}$ Continuous

$X=\{a,b,c\}$ and $\mathcal{T}=\{X, \emptyset, \{a\}, \{b\}, \{a,b\}\}$. Assume $f: X \rightarrow X$ is given by $f(a)=a, f(b)=c,$ and $f(c)=b.$ Determine if $f:X \to X$ is ...
0
votes
1answer
52 views

if $A \subset B \subset X$ then $\mbox{dist}(x,B) \le \mbox{dist}(x,A)$

Prove that if $A \subset B \subset X$ then $\mbox{dist}(x,B) \le \mbox{dist}(x,A)$. My attempt (by contradiction) : Suppose that $\mbox{dist}(x,B) > \mbox{dist}(x,A)$. So we have that: $(\forall ...
1
vote
1answer
167 views

Showing a topology is not metrizable

Show $\prod_{N} \mathbb{R}$ with the box topology is not metrizable. The Box Topology on $\prod_{j \in J} X_j$ ($X_j$ topological spaces) is generated by the basis $\left\{\prod_{j \in J} U_j \; ...
3
votes
2answers
115 views

Relative sizes of Skorokhod and product topologies on space of sample paths

Let $E$ denote a compact metric space. Let $T$ denote the non-negative reals. Let $E^T$ denote the class of all functions from $T$ to $E$. Let $C$ denote the subset of $E^T$ consisting of càdlàg ...
0
votes
1answer
42 views

Open set and sequences in $\mathbb{R}$

Let $A \subset \mathbb{R}$. Prove that $A$ is an open set if, and only if, the following condition is satisfied: " if a sequence $(x_n)$ converges for a point $a \in A$, then $x_n \in A$ for all $n$ ...
1
vote
1answer
36 views

Density of the function class

Let $X$ be any set and let $[0,1]^X$ (the class of all functions $X\to[0,1]$) be endowed with the metric given by $\rho(f,g):=\sup_{x\in X}|f(x) - g(x)|$. Consider any class of functions $\mathscr ...
2
votes
2answers
268 views

Show that $\displaystyle\prod_{\Bbb{N}} \Bbb{R}$ with the box topology is Hausdorff but not metrizable.

Show that $\displaystyle\prod_{\Bbb{N}} \Bbb{R}$ with the box topology is Hausdorff but not metrizable. $\Bbb{R}$ must be Hausdorff. For $x_1, x_2 \in \Bbb{R}$ (where $x_1 \not= x_2$), if $d$ ...
3
votes
2answers
283 views

Infinite disjoint class of open subsets in an infinite Hausdorff space.

Let X be an infinite Hausdorff space. Prove that there exist an infinite disjoint class of open subsets of X. Ok the first time I tried to prove this I started by taking pairwise disjoint sets given ...
2
votes
1answer
63 views

Another question on second countable spaces

Let $X$ have countable chain condition and point countable base. Is $X$ second countable? I thing it don't need to be. However I have no examples at hand. Thanks for your help.
1
vote
2answers
45 views

Show that $\bigcup _{\alpha \in J}\operatorname{cl} A_\alpha \subseteq\operatorname{cl}\bigcup _{\alpha \in J} {A_\alpha} $

$\newcommand{\cl}{\operatorname{cl}}$Let $\{A_\alpha\}_{\alpha\in J}$ be a collection of subsets of a topological space X. Show that $\bigcup _{\alpha \in J}\cl A_\alpha\subseteq\cl\bigcup _{\alpha ...
5
votes
2answers
107 views

Applications of showing a set is both open and closed?

A general technique is as follows: To show that a property holds for a connected space, one can prove that the set of all points that satisfy this property is nonempty and forms a closed and open ...
4
votes
2answers
88 views

What is a 0-ball?

I'm reading a paper that says $\bigcap V_{T,X}$ is either empty or a closed $l$-ball where $T \subset S$ is a subset of points $S$ and $\operatorname{card}{T} = m + 1 - l$ where $m$ is the dimension ...
0
votes
1answer
75 views

A = (0, 1/2). Find the closure of A in X = (0,1].

$A = (0, 1/2)$. Find the closure of $A$ in $X = (0,1]$. So $X = (0,1]$ is a topological space with subspace topology $T' = \{ U \bigcap (0,1] \mid U\text{ is open in }\mathbb{R}\}$. The basis for ...
1
vote
1answer
61 views

Prove that $\bar{h}:Y\to Z$ is a continuous function.

I've run into this rather tricky question (to me at least). Let $(X,\mathcal{T})$ be a topological space and let $Y$ be another set and let $f:X \to Y$ be a surjective function, and equip $Y$ with ...
3
votes
2answers
164 views

Why it does not produce a Klein bottle?

I cannot understand why the action $\mu : (\mathbb{Z}\oplus \mathbb{Z})\times \mathbb{R}^2 \longrightarrow \mathbb{R}^2 $ given by $\mu((m,n), (x, y)) = (x+ m, (-1)^m(y + n))$ does not produce the ...
2
votes
2answers
202 views

How to construct a contractible space but not locally path connected?

I am looking for a space which is contractible and not locally path connected. I know the cone $CX$ of every space $X$ is contractible. Besides, it seems that if $X$ is locally path connected, so is ...
1
vote
2answers
68 views

Proving that $\Bbb{R}_\ell$ is finer than $\Bbb{R}$.

Let us take the two topologies $\Bbb{R}_\ell$ and $\Bbb{R}$. The book "General Topology" by Munkres says that $\Bbb{R}_\ell$ is finer than $\Bbb{R}$. This article says that every open set of $\Bbb{R}$ ...
1
vote
3answers
148 views

Is the set of extended real-valued numbers open or closed

If I assume that my topology is defined on the extended real-valued numbers, then $\mathbb{R}\cup\left\{-\infty,+\infty\right\}=\left[-\infty,+\infty\right]$, acting as my entire space, is both open ...
0
votes
2answers
58 views

Is this a valid proof that $\mathbb{R}$ is connected?

Suppose $\mathbb{R}$ is not connected, i.e., $\mathbb{R}=A \cup B$, where $A,B$ are open sets that are disjoint. $A$ is bounded above by each element of $B$, so $A$ must have a supremum, call it $x$. ...
1
vote
2answers
576 views

Difference between interior and set of accumulation points

I don't understand the difference between the interior of a set, and the set of all its accumulation points. My understanding of an accumulation point is any point in a set which has an epsilon ...
3
votes
1answer
71 views

$A$ is uncountable $\implies$ $A'$ is uncountable?

For $A⊆\mathbb R$ , let $A'$ denote the set of all limit points of $A$ . If $A$ is uncountable , then does it necessarily mean that $A'$ is also uncountable ?
0
votes
1answer
217 views

Proof: A connected metric space which contains more than 1 point is never countable. [duplicate]

This is an exercise in Munkres's book of topology. If $X$ is a connected metric space and there are at least two points in $X$, then $X$ is not countable. I have attempted to find the proof by ...
-1
votes
2answers
67 views

A question on second countable space

A family $\mathcal U$ of subsets of a space $X$ is called k-in-countable if every set $A \subset X$ with $|A|=k$ is contained in at most countably many elements of $\mathcal U$. If $X$ is a ...
0
votes
2answers
75 views

Prove that a continuous $f$ in $(0,1)$ can be extended into its one-point compactification if the limit at both end point exist and equal

Let $X = (0,1)$. Consider the one-point compactification of $X$ (which is homeomorphism to $S^{1}$). Prove that a bounded continuous function $f:(0,1) \rightarrow R$ is extendable to this ...
3
votes
2answers
276 views

Questions on positive definite matrices

First, in this discussion, I am only considering real matrices. Second, I have a few questions I am ruminating on related to symmetric matrices. Some of these questions I need someone to say my ...
0
votes
1answer
73 views

Baire Category Theorem in a Smooth Manifold

Let $Z\subset M$ be a set of measure-0 , in a [smooth] manifold $M$. How does one shows that $M$ \ $Z$ is everywhere dense in $M$, using Baire category theorem? and which of the theorem version is ...
1
vote
1answer
698 views

Let X be any uncountable set with the cofinite topology. Answer the three questions:

Let X be any uncountable set with the cofinite topology. Is this space 1st countable? I don't think this space is 1st countable because it seems that there must be an uncountable number of ...
1
vote
2answers
215 views

Need an example of a space which is not first countable

Give an example of a space which is NOT first countable & in which every singleton set is : $ G_\delta $ . I have just found out one example where the space is NOT first countable is: any ...
2
votes
2answers
232 views

Prove that the identity map $(C[0,1],d_1) \rightarrow (C[0,1],d_\infty)$ is not continuous

$$d_\infty = \max|x_i - y_i|$$ $$d_1 = \sum_{i=1}^n |x_i - y_i|$$ The first part of this question was to prove that the identity map $$(C[0,1],d_\infty) \rightarrow (C[0,1],d_1)$$ is continuous, ...
0
votes
1answer
56 views

Connectedness of subsets

Let $(X,d_x)$ be a metric space, and let $A$ be a non connected, non empty, closed subset of $X$. Can I conclude that $A$ equals to $X$ and $X$ is not connected? It seems that I had wrong ...
1
vote
1answer
55 views

If $F_1 \cup F_2 = \mathbb{R}$ and $F_1,F_2$ are closed sets then $\mbox{Int}F_1 \neq \emptyset$ or $\mbox{Int}F_2 \neq \emptyset$

Prove that if $F_1 \cup F_2 = \mathbb{R}$ and $F_1,F_2$ are closed sets (in euclidean space) then $\mbox{Int}F_1 \neq \emptyset$ or $\mbox{Int}F_2 \neq \emptyset$ My idea is prove that by ...
2
votes
2answers
544 views

Show that the topological space ( X, $\tau$ ) is not metrizable

For the topological space ( X, $\tau$ ), with X = {0, 1} and $\tau$ = { $\emptyset$ , {0}, {0,1} } , prove that ( X, $\tau$ ) is not metrizable. I know intuitively it can't be but don't know how to ...
1
vote
1answer
30 views

Where I can find reference on Katetov's extension $kN$ of the natural numbers?

I’m looking for references on Katetov's extension $kN$ of the natural numbers? However I cannot find it. Is this separable and is this a countable union of closed discrete subspace of it? Thanks ...
2
votes
4answers
117 views

Is $\left\{ \frac{1}{n}: n \in \mathbb{N} \right\} \cup \left\{ 0\right\}$ closed set? [duplicate]

Is it true that $\left\{ \frac{1}{n}: n \in \mathbb{N} \right\} \cup \left\{ 0\right\}$ is closed set? I suppose that yes, but I have no idea how can I prove it.
0
votes
1answer
31 views

Questions on covering space

Following is a paragraph of a paper I am reading: But I cannot understand this image, maybe it is because I have no idea about coverings. Could anyone explain it to me? Particularly, you could just ...
0
votes
2answers
56 views

what is $X / \cong $ ?? where $\cong $ is given by

what is $X / \cong $ ?? Suppose $X = \mathbb{R}^2 $. and we define $$ (x_1,y_1) \cong (x_2, y_2) \iff x_1 + y_1 =x_2 + y_2 $$ With this equivalence relation, we get that the partition is the ...
4
votes
0answers
115 views

Topologist Sine Curve

I am trying to prove that the topologist sine curve is not path connected. I think I have a proof but my proof relies on Intermediate Value Theorem. So, I was wondering if there is a way to prove it ...
0
votes
1answer
218 views

The intersection of open intervals.

For $i=1,2, \cdots, n$, let $I_{i}=(a_i, b_i)$ be an open interval. Show that $\cap_{i=1}^n I_i$ is either the empty set or an open interval. Can anyone show me how to do this because this is slightly ...
0
votes
4answers
171 views

A base for a topology

I am quite confused about what exactly a base for a topology is. I understand it when the topologies are pretty simple, but things start to get a little confusing for me after awhile. For example, ...
0
votes
2answers
49 views

One question about topology.

Why is the set $ A=\{ (x ,x^{-1}):0<x\leqslant 1\}$ is closed in $\Bbb R^2$ but is not bounded? Why is the set $ S=\{(x,\sin(x^{-1})) :0<x\leqslant 1\}$ is bounded in $\Bbb R^2$ but is not ...
4
votes
1answer
366 views

Non-Euclidean Space in Dungeons and Dragons

In Dungeons and Dragons, the world is mapped out into five-foot squares. Spheres are represented as cubes, and cones look really weird. However, straight lines remain straight, and a rectangular room ...