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 (3)

2
votes
2answers
20 views

Locally compact Hausdorff space and indicators

This is exercise 6 from Tao's notes on locally compact Hausdorff spaces. Let $X$ be such a space and assume $K \subset U$ where $K$ compact and $U$ open. We want to find a function $f:X \to \mathbb ...
0
votes
2answers
45 views

Show that if A is bounded above, then it contains its supremum…

Please check my answer. ..........Question......... Suppose $A \subseteq \mathbb{R}$ is closed and nonempty. Show that if $A$ is bounded above, then it contains its supremum, and if it is bounded ...
1
vote
2answers
51 views

If $A\subseteq B\subseteq\mathbb{R}^{2}$, $A$ open, $B$ closed, then $\overline{A}\subseteq B$?

If $A\subseteq B\subseteq\mathbb{R}^{2}$, $A$ open, $B$ closed, then $\overline{A}\subseteq B$? That is, $\partial A \subseteq B$?
1
vote
0answers
37 views

Are these subsets open, closed, both or neither (revised)?

This is a follow up to Are these subsets open, closed, both or neither? Please let me know if my answers are correct, and If my reasoning is accurate and complete. Below are my corrections: ...
0
votes
3answers
64 views

If a mapping $g: \mathbb{R} \to \mathbb{R}$ is strictly increasing, is it an open map?

If a mapping $g: \mathbb{R} \to \mathbb{R}$ is strictly increasing, is it necessarily an open map? i.e. for $a,b \in \mathbb{R}$ and $a<b$ can we conclude that ...
2
votes
1answer
60 views

Open cover of non-compact spaces

Let $X$ be a non-compact space. (A space is compact if any open cover has a finite subcover.) I want to show that there is an ordinal $\alpha$ and an open cover $(U_\xi)_{\xi < \alpha}$ such that ...
0
votes
0answers
43 views

General product of homeomorphic spaces

I came across this question in general topology and I genuinely am baffled. If somebody could give me an answer, or point me in the right direction, it would be massively appreciated. Suppose we ...
2
votes
1answer
53 views

Topology question with closed sets.

Let $ K\subseteq \mathbb{R}^n$ be a compact set and let $E\subseteq \mathbb{R}^n$ be a closed set. ***Its also given that $ \inf \{d(x,y)|x\in K, y\in E\}=0$. $ d(x,y)=\sqrt{\sum_j (x_j-y_j)^2}$ ...
2
votes
1answer
31 views

Irregular (branched) cover

I need to know the definition of an irregular (branched) cover. I heard this somewhere but I am not able to find any definition on the internet.
7
votes
6answers
884 views

If a nonempty set of real numbers is open and closed, is it $\mathbb{R}$? Why/Why not?

If a nonempty set of real numbers is open and closed, $\mathbb{R}$ Why/Why not? In other words, are $\emptyset$ and $\mathbb{R}$ the only open and closed sets in $\mathbb{R}$? Why/Why not? I tried ...
6
votes
3answers
185 views

What is the sheafification of the presheaf of the one point compactification?

Okay, so I had this idea for a presheaf that is quite peculiar. Instead of being based on algebraic category (i.e. abelian groups), it is based on a topological one, the category of compact ...
7
votes
4answers
270 views

Simple question on closed sets

A closed set is one which contains all its limit points. Why is $[a, \infty)$ closed? Specifically I don't understand how $\infty$ which is a limit point, but it is not in the set.
11
votes
1answer
105 views

If every point is a local maximum, is it a step function?

What are the functions $f:\mathbb R\to\mathbb R$ such that every point is a local maximum? Certainly, $f(x)=c$ works for every constant. So does $\lfloor x\rfloor$, as does ...
0
votes
0answers
45 views

Info on the locale of surjections from the Natural Numbers to the Real Numbers

On the nlab page for locales, it states that there is locale for the surjections from the Naturals to the Reals. This locale has no points (i.e. elements), since there are no such surjections, but the ...
3
votes
2answers
123 views

Slick proof that if an open set contains $\mathbb Q$ it has all irrational numbers, except a countable amount.

Basically I need help in proving that if $U\supseteq \mathbb Q $ is an open set in $\mathbb R$ with the usual topology then $\mathbb R \setminus U$ is countable. I'm not really sure how to proceed. ...
5
votes
2answers
2k views

Proof that every metric space is homeomorphic to a bounded metric space

I have tried to show that every metric space $(X,d)$ is homeomorphic to a bounded metric space. My book gives the hint to use a metric $d'(x,y)=\mbox{min}\{1,d(x,y)\}$. If we can show that $d(x,y) ...
1
vote
1answer
17 views

Local extension of a function on an immersed submanifold

Consider the following passage in Spivak's Differential Geometry book: I am having trouble understanding where he says $g = \tilde{g} \circ i$ on $V \cap M_1$. Since $V$ is (I think) supposed to be ...
72
votes
0answers
2k views

Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
6
votes
1answer
72 views

$R^2$ is not isometric to $R^3$

Is there a direct proof for showing that $R^2$ is not isometric to $R^3$ (with the usual metrics)? I know that they are not homeomorohic but I think there should be some direct and easy proof for ...
3
votes
3answers
81 views

Extension of metric definition to two sets

The standard definition of a metric, is a function $d: X\times X \to \mathbb{R}$. What is a sensible/common extension of a metric/pseudometric to $\tilde d: X\times Y \to \mathbb{R}$, i.e. distance ...
4
votes
1answer
672 views

When is the closure of a path connected set also path connected?

What are the most general criteria we can impose on a locally path connected Hausdorff space $X$ and a path connected subset $A$ such that $\overline{A}$ is path connected? Do more restrictions need ...
-4
votes
2answers
72 views

Topological properties that the real line does not have

The following question is kind of strange, but I would like to know what topological properties $\mathbb{R}$ (with the standard metric topology) does not posses? I know this question sounds a bit ...
1
vote
1answer
44 views

Are these subsets open, closed, both or neither?

I'm teaching myself topology using a text I found online. Right now I'm reviewing "Metrics." Please let me know if my answers are correct, and If my reasoning is accurate and complete. I think (c)and ...
2
votes
1answer
65 views

What does it mean for a topological space to contain a copy of $\mathbb{R}$?

If $X$ is a topological space equipped with topology $\tau$. I have come across some concept while going through some of the content related to topology. It says, "The topological space $X$ contains ...
12
votes
7answers
290 views

Moebius band not homeomorphic to Cylinder.

I have been trying to think of a rather basic way of proving this, but it seems a bit elusive. In the case with boundary (looking at them as quotients in $ [0,1] \times [0,1] $), they can be ...
4
votes
3answers
107 views

Is the product of a proximal system with itself proximal?

A topological dynamical system is a pair $(X,T)$ where $X$ is a compact metric space and $T$ is a continuous map from $X$ to itself. Two points $x,y\in X$ are said to to be proximal if for any ...
2
votes
1answer
61 views

Does trivial fundamental group imply contractible?

Let $X$ be a path-connected topological space with a trivial fundamental group: $$\pi_1(X,x_0)=\{e\}.$$ Does $X$ have to be homotopic to a point? I know that the converse is true: a ...
1
vote
0answers
41 views

Closed unit ball is a retract of $R^2$

I was asked whether a closed unit ball is a retract of the euclidean space $R^2$. I think the answer is yes and the retraction might be defined as follows: for all the points in $R^2$ join them with ...
1
vote
2answers
30 views

Does finite covering dimension imply local compactness?

I have a space which is not locally compact and I'm trying to see if I can say anything about the dimension of the space. I suspect that it is not finite dimensional but I have thus far been unable to ...
3
votes
2answers
52 views

Help finding the fundamental group of $S^2 \cup \{xyz=0\}$

let $X=S^2 \cup \{xyz=0\}\subset\mathbb{R}^3$ be the union of the unit sphere with the 3 coordinate planes. I'd like to find the fundamental group of $X$. These are my ideas: I think the first thing ...
0
votes
1answer
36 views

Why is the set composed of all intervals $(-r,r)$ for any rational number $r$, the empty set, and all real numbers not a topology?

It seems like it ought to be:- $(-a,a) \cap (-b,b)$ where $a \leq b$ is $(-a,a)$ and $(-a,a) \cup (-b,b)$ is $(-b,b)$, so it should be closed under union and intersection. What am I missing?
0
votes
1answer
38 views

Alternative Proof of why Every Manifold is Locally Compact

So while I was solving some problems on differential geometry, I stumbled upon a problem which is to show that every manifold is locally compact. Now, there is a proof for it here, but I was thinking ...
2
votes
0answers
30 views

Discrete closed subsets in Lindelöf spaces

Let $X$ be a Lindelöf space and let $A$ be a discrete closed subset of $X$. Then $A$ must be countable. Indeed, for any $x \in A$, the singleton $\{x\}$ is an open subset of $A$. Obviously, the sets ...
0
votes
0answers
30 views

A detail in the proof orientable manifold admits exactly two orientations

Let $M$ be a real manifold and let $\{(U_\alpha,\phi_{\alpha})\}_{\alpha\in I}$ $\{(V_\beta,\psi_\beta)\}_{\beta\in J}$ be two oriented atlases. Let's define, for $p\in U_\alpha\cap V_\beta$, ...
1
vote
2answers
50 views

Two definitions of one set being dense in the other

Assume $\langle X,\mathscr{O}\rangle$ is a topological space. Let $A,S\subseteq X$. Let $\langle S,\mathscr{O}\rangle$ be a subspace of $X$ with $\mathrm{Cl}_S$ being its closure operation. There are ...
0
votes
1answer
54 views

Point sets and limit points.

Show that if M is a point set having a limit point, then M contains at least 2 points. Must M contain 3 points? 4 points? Having difficulty describing and visualizing, because it seems rather ...
2
votes
2answers
42 views

Open balls with radis $>\epsilon$ in a compact metric space

In a compact metric space $(X,d)$, for a given $\epsilon>0$, if $(x_j)_{j \in J}$ is a family of points of $X$ such that the balls $B(x_j, \epsilon)$ are pairwise disjoint, does it automatically ...
20
votes
7answers
2k views

Perfect set without rationals

Give an example of a perfect set in $\mathbb R^n$ that does not contain any of the rationals. (Or prove that it does not exist).
0
votes
1answer
76 views

Show that $S$ is connected

Let $S=\{x\in\mathbb R^n:||x||=1\}$ with $ n>1$. Show $S$ is connected without using arcwise connectedness. I would be done if I can show this: Let $X$ be a connected space and $A\subset X$ be ...
2
votes
1answer
200 views

Path connected attaching map

I am trying to showed that if $X$ and $Y$ are path connected then $X \coprod_f Y$ is path connected. (The adjunction space). Let $A \subset X$ , and let $f:A \to Y$ be the attaching map. (Note: ...
0
votes
2answers
77 views

Why are + and $\times$ considered continuous on a topological vector space? [closed]

$\mathbb{R}$ is a topological vector space. Elements of $\mathbb{R}$ are real numbers. Every number on topological vector space is closed. A function $f: X \to Y$ is considered continuous if every ...
2
votes
1answer
104 views

Can the same subset be both open and closed?

This is a follow up response to: Counterexample to " a closed ball in M is a closed subset." I'm trying to understand this using only the given definitions of: Metric space, open/closed ...
0
votes
0answers
51 views

Topological cardinal function : The grasp

Def.: For a topological space $S$ with weight $w(S)=\kappa$, I define the grasp $g(S)$ to be the least infinite cardinal $\gamma$ such that $S$ has a base $\mathscr{B}$ with $|\mathscr{B}| \le \kappa$ ...
3
votes
0answers
78 views

Classify knots in a closed bead-spring like polymer simulation

I'm trying to detect the crossing number (or another knot invariant) of a simulated polymer. A polymer is a closed bead spring, which means that it is represented by a set of points connected by ...
4
votes
5answers
718 views

Show that $X=\{0\} \cup \{\tfrac{1}{n}:n \in \mathbb{Z} - \{0\}\}$ is a complete metric space

How could we show that the metric space $$X=\{0\} \cup \{\frac{1}{n}:n \in \mathbb{Z} - \{0\}\}$$ with the metric it inherits as a subset of $\mathbb{R}$ is complete? Thoughts Complete metric ...
0
votes
1answer
34 views

Trouble understanding Banach limit in Stone-Cech compactification of $\mathbb{N}$

Trouble understanding Banach limit in Stone-Cech compactification of $\mathbb{N}$. For example, if I have a series $\{a_n\}_{n\in \mathbb{N}} \in [0,1]$, what does it mean that the limit of the series ...
18
votes
5answers
1k views

Counterexample to “ a closed ball in M is a closed subset.”

I am studying topology, on my own, using a text I found online. I am currently reviewing the “Metrics” section that reminds me of the real analysis course I took over 10 years ago. The text ask me to ...
0
votes
0answers
27 views

Why doesn't the proof of the Urysohn lemma go through if we have a regular space only?

I'm reading James Munkres. The author writes, page 211, that take a point $a$ and a closed set $B$ not containing $a$. Define $U_1 = X - B$ and choose the open set $U_0$ about $a$ whose closure is ...
0
votes
0answers
17 views

Help with the proof of the Urysohn metrization theorem in James Munkres's Topology, page 215.

I don't understand how we're able to apply the Urysohn lemma in Step 1 when the given space $X$ is only regular, and not normal.
-5
votes
1answer
33 views

it is a problem on topology [closed]

let $X$ BE LOCALLY COMPACT SECOND countable hausdorff space show that there exist a sequence {$K_n$} of compact subset such that $X=\cup K_n$ and $K_n \subset Int(K_{n+1}) $ .