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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2
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1answer
97 views

Topology(meaning) [closed]

When we define Topology we say that a topology on a set(let's say X) is a collection of subsets of X having certain 3 properties. Now, here what do we actually mean by saying "topology on a set". What ...
2
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1answer
52 views

Image of isometric immersion

Let $M$ a metric space with the following property: For all isometric immersion $f:M\to N$, the image of $f(M)$ is a open set in $N$. Prove that $M$ is empty set. A function $f:M\to N$, is called ...
0
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2answers
62 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 ...
2
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2answers
31 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 ...
3
votes
1answer
58 views

Is $f(x,y)=ax^2+by^2, \ a,b \in \mathbb R $ a bijection between $\mathbb R^2 \to \mathbb R$? Bijections of topologies

Is $f(x,y)=ax^2+by^2$ a bijection between $\mathbb R^2 \to \mathbb R$ ? How about $f(x,y,z)=\frac{x^2}{a^2} + \frac{y^2}{b^2}+ \frac{z^2}{c^2}? ( \mathbb R^3 \to \mathbb R )$ What confuses me now ...
1
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2answers
65 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
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0answers
69 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
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3answers
70 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 ...
3
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2answers
89 views

Is there a cute proof of (Hausdorff space iff diagonal closed)?

Let $(X,\tau)$ be a topological space, and let $\Delta_{X}=\{(x,x)\mid x \in X\}$ be its diagonal. It is known that $\Delta_X$ is closed if and only if $(X,\tau)$ is Hausdorff. I know how to prove ...
1
vote
1answer
313 views

Parallelizability of Lie groups

I have tried a lot to prove this well-known result. The basic idea behind the proof is clear to me. But I'm stuck at showing either of the following conditions: The map $\ G \rightarrow TG \ $ given ...
2
votes
1answer
63 views

Confusion regarding the $\omega$-limit of a set in a flow

In Salamon's Connected Simple Systems, p.8, the author writes that the $\omega$-limit of a set $Y$ inside a flow $\Gamma$ has the two equivalent descriptions $$ \omega(Y) = I(\overline{Y \cdot ...
3
votes
1answer
36 views

Can $S^4$ be the cotangent bundle of a manifold?

I am asking the question because in the classical mechanics book by Arnold, he states that there is a distinguished 1-form on $T^*V $. It seems that there is no such distinguished 1-form on a general ...
2
votes
1answer
66 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
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0answers
46 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
57 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}$ ...
7
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2answers
214 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 ...
1
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0answers
63 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 ...
7
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4answers
304 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.
3
votes
2answers
155 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. ...
11
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1answer
178 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 ...
2
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1answer
43 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.
2
votes
1answer
87 views

Show that every open subset of a locally connected space is locally connected.

Suppose $X$ is a topological space that is locally conneted and let $O$ be an open subset of $X$. Then we want to show that $O$ is also locally connected. Let $p\in O$ chosen arbitrarily, then there ...
1
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1answer
157 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
42 views

Classification of dense and complete linear orders

Question. Is there a decent classification theorem for linear orders satisfying all three of: Dense. Given a pair of elements $y,x$ with $y>x$, there exists $k$ satisfying $y>k>x$. ...
6
votes
1answer
88 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 ...
2
votes
1answer
117 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
59 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 ...
-2
votes
2answers
88 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 ...
3
votes
3answers
97 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 ...
3
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1answer
62 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
42 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?
2
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0answers
33 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
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1answer
52 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 ...
0
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0answers
45 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 $p\in U_\alpha\cap V_\beta$, ...
2
votes
1answer
70 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 ...
2
votes
2answers
88 views

Algebra formed by operators and Kuratowski's theorem

I have been reading the paper "D. Sherman, Variations on Kuratowski's 14-set theorem, Amer. Math. Monthly 117 (2010), no. 2, 113-123" recently. Kuratowski Closure complement theorem states that: Let ...
1
vote
1answer
24 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 ...
0
votes
1answer
87 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 ...
1
vote
2answers
47 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 ...
1
vote
2answers
82 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 ...
0
votes
1answer
92 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 ...
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0answers
110 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
1answer
155 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 ...
2
votes
1answer
178 views

Density of subspace with nonlocal/Wentzell boundary condition

Given the space $F$ defined by: $$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$ I want to prove that the subspace $E$ of $F$ defined by ...
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2answers
85 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
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0answers
34 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 ...
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0answers
36 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.
1
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1answer
36 views

Why does the Hausdorff metric need to be defined on bounded subsets only?

Claim: Suppose $X$ is a non-empty set and $d$ is a metric on $X$. Let $S(X)$ denote the collection of all non-empty closed bounded subsets of $X$. For each $A$ and $B$ in $S(X)$, define $$h(A,B) = ...
0
votes
1answer
17 views

Why $F_{|X\times {\{t}\}}$ for all $t\in [0,1]$ is homeomorphism?

The following text is the definition and an example for isotopy: I don't understand why $F_{|X\times {\{t}\}}$ for all $t\in [0,1]$ is homeomorphism. According to the definition of homeomorphism, ...
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votes
1answer
36 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}) $ .