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
126 views

Proof of the Inverse Function Theorem using the Contraction Mapping Principle.

I've been set this problem recently and I'm having a lot of trouble with it. Any help would be much appreciated! Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function with continuous derivatives ...
-1
votes
3answers
93 views

Is this true: Every open set $A$ contains a neighborhood whose closure is a subset of $A$. [closed]

This seems a very easy fact. But I don't know how to prove it. Can anybody help me? Thanks!
1
vote
2answers
56 views

Critique my proof from munkres product topologies?

I've been going through Munkres' book on topology on my own, and I just struggled through the proof of 10d) from chapter 2 section 19. I've never had a chance to show one of my proofs to anyone, so I ...
0
votes
1answer
27 views

Conjecture about regular Borel measures and dense sets with no interior

Suppose that $(X,\tau)$ is a topological space and let $\mathscr B$ denote the Borel $\sigma$-algebra on it. Moreover, let $\mu:\mathscr B\to[0,\infty]$ be a regular Borel measure, that is, ...
4
votes
1answer
111 views
+50

Metric limit and limit in category

Is it possible to construct a category $\mathcal{C}$ with $\mathrm{Ob}\,\mathcal{C}=\mathbb{R}$ and for every diagram of the from $$a_0\leftarrow a_1\leftarrow\cdots a_n\leftarrow\cdots$$ the inverse ...
0
votes
1answer
30 views

Simlpe Loops in Topological Graph

Given a set of points in 3D space, and a set of links between them which form a connected graph - is there a general strategy for extracting all simple loops from such an object? I refer to simple ...
6
votes
1answer
32 views

A proper subspace of a normed vector space has empty interior.

In a vector normed space $E$, prove that all vectorial subspace $F\neq E$ has a interior empty. My approach:We consider, the open ball $B\subset F$, with $F$ proper subspace of $E$. If $x\notin F$, ...
0
votes
3answers
91 views

Definition of compactness unnecessarily verbose?

The definition of a compact set is given as a set, $X$, for which all open covers have a finite subcover. This seems unnecessarily verbose to me. Wouldn't it be sufficient to simply say that $X$ has ...
0
votes
1answer
21 views

Is a limit point compact subset of a Hausdorff space necessarily closed?

This is Exercise 3 c) from Section 28 of Munkres - Topology. I had thought the answer was no initially, but the example I came up with was not Hausdorff. At this point, I'm just not sure. Here's ...
2
votes
1answer
54 views

Prove there exists a unique local inverse.

I've been set this problem recently and I'm having a lot of trouble with it. Any help would be much appreciated! Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function with continuous derivatives ...
5
votes
1answer
55 views

Continuity of a “minimal distance” projection $f:(X,d) \to (K, d_{|K})$ for a compact $K \subset X$. (Hint preferred)

Let $(X,d)$ be a metric space and $K$ be a compact subset of $X$. Show that for every $x \in X$ there exists $k_x \in K$ such that $$d(x,K)=d(x,k_x)$$ Suppose that for every $x\in X$, there exists ...
0
votes
0answers
14 views

Are some some particular subspaces of cadlag functions Polish?

Consider the space $D := D((0, \infty), \mathbb{N})$ of cadlag functions $f : (0, \infty) \to \mathbb{N}$ equipped with the Skorokhod $M_1$-topology. Then $D$ is Polish. Question 1: I want to check ...
0
votes
1answer
28 views

Proof that a homeomorphic image of a non-borel set is non-borel

This question seeks to expand the proof given in the answer to this question. I am weak in topology, and am wondering if someone can provide a proof of why a homeomorphic image of a non-borel set is ...
-3
votes
4answers
73 views

A compact open set

Is there an open set which is compact ? I would say that $\emptyset$ is an open set compact because it's bounded and closed too. Is it correct ?
3
votes
1answer
55 views

If $U\subseteq V \subseteq \mathbb{R}^{2}$, then $\partial U \cap V =\partial\left(V\setminus U\right)\cap V$?

Assume $U\subseteq V \subseteq \mathbb{R}^{2}$, both open, is then $\partial U \cap V =\partial\left(V\setminus U\right)\cap V$? Edit: What if $U, V$ also are simply connected?
2
votes
1answer
81 views

Manifold is not orientable

Let $M$ be a manifold of dimension $n$ such that there exist two charts $(U_a,\phi_a)$ and $(U_b,\phi_b)$ such that $U_a,U_b$ are connected and $U_a\cap U_b\ne\emptyset$. Moreover the ...
1
vote
1answer
53 views

Countable choice and totally bounded metric spaces

Can we prove that the following statement is equivalent to the axiom of countable choice (CC)? If every sequence in a metric space $X$ has a Cauchy subsequence, then $X$ is totally bounded. ...
0
votes
0answers
45 views

Open Set in the Cartesian plane.

I'm trying to prove that the following set is an open set in $\mathbb{R}^2:$ $$A=\{(x_{1},x_{2})\in\mathbb{R}^{2}: x_{1}+x_{2}>1\}$$ with respect to norm $||x||_{1},||x||_{2},||x||_{\infty}.$ ...
0
votes
1answer
18 views

Is the condition “the inverse image of a closed base set is closed?” sufficient for continuity?

Let's say you have a function $f:X \to Y$, where $X$ and $Y$ have topologies. The set $C$ forms a closed base for $Y$. If for every $c \in C$, $f^{-1}(c)$ is closed in $X$, is $f$ continuous? If the ...
1
vote
1answer
49 views

The empty set is a neighborhood?

The following axioms of a Topological space is from Wikipedia: Neighbourhoods definition This axiomatization is due to Felix Hausdorff. Let $X$ be a set; the elements of $X$ are usually ...
5
votes
2answers
59 views

Prove $\{(x,y): x>0\}$ is connected

As an introduction to multivariable calculus, I'm given a small introduction to some topological terminology and definitions. As the title says, I have to prove that $\{(x,y): x>0\}$ is connected. ...
0
votes
0answers
18 views

Deforming path of integration from the real line to the boundary of a open subset of the upper half complex plane.

Denoted the upper half of the complex plane by $\mathbb{C}^{+}=\{z\in\mathbb{C}:\text{Im }z>0\}$. Let the open, unbounded set $A\subseteq\mathbb{C}^{+}$ have a boundary $\partial A$ such that the ...
2
votes
1answer
71 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
votes
1answer
35 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
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 ...
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 ...
3
votes
1answer
56 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
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 ...
3
votes
2answers
66 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
96 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
36 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 ...
2
votes
1answer
21 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
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}$ ...
6
votes
3answers
184 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 ...
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 ...
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.
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. ...
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 ...
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.
2
votes
1answer
26 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
vote
1answer
43 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
30 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
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 ...
2
votes
1answer
60 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
39 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 ...
-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 ...