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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1
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4answers
27 views

How to determine the closure of a subset and prove it is actually the closure?

I have this subset $E = \{r \in Q: r^2 \leq 2\}$ which is in $\mathbb{R}$ with the Euclidian metric. I was wondering how can I find the closure of this subset. Here is what I have: The limit points ...
1
vote
1answer
58 views

Finite mapping $f : \mathbb R^2 \to \mathbb R$

Is there an continous function $f: \mathbb R^2 \to \mathbb R$ such that $f^{-1}(a)$ is finite for every $a \in \mathbb R$? It's not possible for analytic or smooth but I'm curious about continous ...
12
votes
2answers
150 views

Completion of the real numbers

On the real line $\mathbb{R}$ endowed with euclidean topology i may put different metrics, inducing the same topology, but inducing different completions. For example if one considers the standard ...
3
votes
1answer
32 views

Surjectivity of $\mathcal{id}_{\mathbb{R}^n}+g$ when $g$ is a contraction?

Assume $g:\mathbb{R}^n\longrightarrow \mathbb{R}^n$ is a contraction and consider $h=\mathcal{id}_{\mathbb{R}^n}+g$. The map $h$ is injective. Is it always surjective? My question has the following ...
0
votes
1answer
41 views

What does arbitrary mean in the following context?

I recently encountered a theorem stating, "The arbitrary product of compact spaces is compact". What does arbitrary product mean in this context? Any product of compact spaces? If that is the case why ...
2
votes
0answers
68 views

Distance between sets

Let $K \subset K_1 \subset U \subset \Bbb R^2$, such that $K$ and $K_1$ are compact sets, with $K \subset \mathring {K_1}$, and $U = \mathring U \subsetneq \Bbb R ^2$. If $w \in \partial K_1$ such ...
1
vote
0answers
18 views

What is the formalism for a map that returns the adjacent vertex positions of a given adjacency matrix?

How do I formally denote a map that returns the adjacent vertex positions of a given adjacency matrix? Example: V = undirected adjacency matrix $$ V = \left[ \begin{matrix} v_{1,1} & v_{1,2} ...
1
vote
0answers
51 views

Prove that $x+g$ is homeomorphism

Problem: Assume we have $g:\mathbb{R}^n\longrightarrow \mathbb{R}^n$ of $C^1$ class with derivative bounded uniformly by some constant $M<1$. Consider ...
1
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0answers
11 views

Proposed proof for quasi-metric result

A quasi-metric on a set $X$ is mapping $\rho: X \times X \rightarrow [0, \infty)$ satisfying the following conditions: $\rho(x,y) \geq 0~~\text{and}~~\rho(x,x) = 0;$ $\rho(x,z) \leq \rho(x,y) + ...
2
votes
0answers
22 views

How do I prove the converse of Stone-Weierstrass theorem?

Let $X$ be a locally compact Hausdorff space. Let $\bar \rho$ be the uniform metric on $\mathbb{R}^X$ and $\mathscr{A}$ be an $\mathbb{R}$-subalgebra of $C_0(X,\mathbb{R})$ which is dense in ...
0
votes
0answers
16 views

Outer Regularity of the Lebesgue measure on the Hilbert brick

Is the product measure on the Hilbert brick $I=[0,1]^\mathbb{N}$ outer regular (that is measure of every set is the inf of measures of open sets, containing it)?
0
votes
5answers
55 views

Finite set of points of $R^n$ is compact

In order to show that a finite set of points of $R^n$ is compact, I just need to show that the set is closed and bounded. First of all, since it's a finite set, I can Always pick the greatest ...
3
votes
1answer
35 views

Is this space Hausdorff and are these two spaces homeomorphic?

Let $S^1 = \{(x,y) \in \mathbb{R^2} \ | \ x^2 + y^2 = 1 \}$. We define an equivalence relation $\sim$ on $S^1$ such that $(x,y) \sim (x',y')$ if and only if $y = y'$. Now we study the following ...
0
votes
0answers
13 views

Cartesian product between manifolds

I was given the following exercise: Show that if $M$ is a $k$-manifold without boundary in $\mathbb{R}^m$, and if $N$ is an $l$-manifold in $\mathbb{R}^n$, then $M \times N$ is a $k+l$ manifold in ...
2
votes
1answer
23 views

Kelly's topology: Compact spaces and finite intersection property

Lemma A topological space is compact iff each family of closed sets which have the finite intersection propertyhas a non-void intersection. I've proved the same result in another way but i really ...
0
votes
0answers
30 views

When is a connected space, path connected?

Let $X$ be a connected topological space. When is $X$ path connected? Is the Hausdorff property enough? Is it too much?
0
votes
1answer
18 views

Prove that a complete field defines a partition of a set

Let $\Omega$ be arbitrary set. Let $Q$ be a partition of $\Omega$. I already proved that the collection of all unions of the cells in $Q$ is a complete field $\mathcal{F}$ (complete field is ...
0
votes
2answers
16 views

limit points of subset of real numbers

Let $$A=\{ \frac{\sqrt{m} -\sqrt{n}}{\sqrt{m}+\sqrt{n}} | m,n\in \Bbb{N} \}$$ I think that we must find sequence of $A$ and find limit of sequence,let $a_m =\frac{\sqrt{ k ^2 m^2} -\sqrt{ ...
1
vote
2answers
42 views

Finding limit points for these sets

Here's my resoning for finding limit points for some sets. Could you guys read it and see if it's all good? <3 $$\{(x,y)\mid \ x^2+y^2<1\}$$ For this set, its kinda simple to see that every ...
1
vote
3answers
53 views

Discontinuous maps taking compacts to compacts

It's commonly known that in general topology, a continuous map $f$ from a topological space $(X, \tau)$ to another topological space $(Y, \tau')$ will send every compact set to another compact set. ...
0
votes
0answers
59 views

Show that if $f$ is differentiable as to function $x\mapsto ||x||$ with $x\in R$,then $f'(0)=0$ [on hold]

Let $f\in C^{\infty}(Ω)$ for some open set $Ω \subset R^n$ that contains $0$. Show that if $f$ is differentiable as to function $x\mapsto ||x||$ with $x\in R$,then $f'(0)=0$. I found this problem in a ...
4
votes
0answers
39 views

Map of smooth manifolds

Let $M$ and $N$ be smooth, connected $n$-dimensional manifolds. Let $M$ be compact and non-empty. Show that every embedding $f: M \to N$ is a diffeomorphism. So because $f$ is a embedding we have ...
0
votes
0answers
34 views

Topology problem [duplicate]

It is prob.5(a) of section 16, ch2 in munkres: Let X and X' denote a single set in the topologies T and T',respectively. Let Y and Y' denote a single set in the topologies U and U', respectively. ...
2
votes
6answers
64 views

Prove that $\{(x,y)\mid xy>0\}$ is open

I need to prove this using open balls. So the general idea is to construct a open ball around a point of the set. A point $(x,y)$ such that $xy>0$. Then we must prove that this ball is inside the ...
1
vote
1answer
27 views

LCA - groups under continuous homomorphisms

can someone help me out with this question? LCA stands for Locally compact Hausdorff abelian group. The question is posted in the attached image Let $A$ and $B$ be LCA-groups and $H$ a (not ...
1
vote
2answers
60 views

Are continuous functions with compact support bounded?

While studying measure theory I came across the following fact: $\mathcal{K}(X) \subset C_b(X)$ (meaning the continuous functions with compact support are a subset of the bounded continuous ...
0
votes
2answers
26 views

Equivalence to being a topological group

Just some notation I am using: A topological group $G$ is a group with a topology such that $o : G^2 \to G : (x,y) \mapsto xy$ and $inv : G \to G : x \mapsto x^{-1}$ are continuous in the ...
0
votes
2answers
42 views

Prove $f$ is Lipschitz on $K$

Let $f:\mathbb{R}^d\to \mathbb{R}$ such that it's partial derivatives are continuous. Let $K\subseteq \mathbb{R}^d$, a bounded set. Prove that $f$ is Lipschitz on $K$. My work: Since $f$'s ...
-4
votes
0answers
33 views

What is the definition of the following concepts and how I can characterize each concept. [on hold]

What is the definition of the following concepts and how I can characterize each concept. A set $A\subseteq 2^{\omega}$ is Lebesgue measurable zero if ? A set $A\subseteq \omega^{\omega}$ is ...
3
votes
0answers
35 views

Fibre of a local homeomorphism can be covered by disjoint open sets.

Let $f\colon X\rightarrow Y$ be an open local homeomorphism and $y\in Y$. Do there exist pairwise disjoint open neighborhoods $U_x$ for $x\in f^{-1}(y)$? If not, what would be mild topological ...
2
votes
1answer
42 views

Countable vector space of continuous functions over a compact metric space

In a proof of a specific theorem, the following is stated: ($\Omega$ is assumed to be a compact metric space) "Let $H \subset C(\Omega)$ be a countable vector space over $\mathbb{Q}$ which is closed ...
2
votes
1answer
19 views

Hausdorff Lindolef Space is Regluar?

I think we can use same argument for saying regluar Lindolef space is normal to prove Hausdorff Lindolef space is regluar. But, I didn't heard about this proposition. What is the problem of using same ...
1
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0answers
30 views

a question about Fenchel's theorem(differential geometry)

I am an undergraduate student studying differential geometry right now. I am just finishing reading how to prove Fenchel's theorem:The total curvature of a smooth closed curve in 3-dimensional space ...
2
votes
2answers
28 views

Compact sets are bounded: shape of the cover matters?

To prove a compact sets is bounded, we assume there's a "open ball cover" (each with R=1) that covers the set. And take maximum distance over the center of the balls +2 as the boundary. Why could we ...
1
vote
3answers
57 views

$x^2+y^2<1, x+y<3$ is open or closed?

I'm trying to figure out if $$\{x^2+y^2<1, x+y<3|(x,y)\in \mathbb R^2\}$$ is open or closed. I tried to imagine this set. It looks, for me, as a 'pizza', or a circular sector, which have two ...
3
votes
1answer
51 views

Open sets in the product topology

Let $\{X_{\alpha}\}_{\alpha\in A}$ be a family of topological spaces. The product topology on $X=\prod_{\alpha\in A}X_{\alpha}$ is the weak topology generated by the coordinate maps ...
2
votes
0answers
47 views

Fundamental group two torus minus a single point?

So, if I take one torus and take of one single point, what will be its fundamental group? I think that one single point will not change the topology in this sense. Or will? If yes, how?
1
vote
1answer
27 views

Show that a regular space, under a “new” topology, is Tychonoff

I have this problem that I can't solve it: Problem: Let $(X, \tau)$ be a regular space. Let $\tau_\delta = \{A \subseteq X \, \: \, \forall a \in A \, \, \exists \, W \subseteq X \textrm{ such that } ...
0
votes
0answers
52 views

Continuous function $0$ on one closed set and $1$ on the other

Looking for a better approach of the following question if possible. Question: Let $A$ and $B$ be disjoint nonempty closed sets in a metric spaces $X$, and define ...
7
votes
1answer
71 views

Limit of measurable functions is measurable?

Suppose $(\Omega, \cal F)$ is a measurable space and $(X, \mathcal B_X)$ is a topological space with its Borel sigma algebra. If $f_n: \Omega \to X$ is a sequence of $(\cal F , B$$_X)$-measurable ...
2
votes
1answer
39 views

Is this argument invalid?

So, what I was trying to prove is: Let $\pi : X \to Y$ the quotient map such that $\pi^{-1}(\{y\})$ connected for all $y \in Y$. Then $X$ is connected. Suppose yet that $X$ is locally connected. ...
1
vote
1answer
52 views

Why is $f(x,y)$ said to be discontinuous at $(0,0)$?

Why is $f(x,y)=\begin{cases} \frac{x^2y}{x^4+y^2}, & \text{if $(x,y)\neq (0,0)$}\\[2ex] 0, & \text{if $(x,y)=(0,0)$} \end{cases}$ said to be discontinuous at $(0,0)$? I am supposed to show ...
0
votes
1answer
28 views

Open (closed) sets of a locally compact space.

Let $X$ a locally compact space. How do I show that if $A$ is a open (closed) set in $X$ then $A$ is locally compact? Thank you very much.
2
votes
3answers
35 views

If $K\in \Bbb{R}^n$ is compact, $\sup_{x,y\in K}|x-y|=\max_{x,y\in K}|x-y|$.

Suppose $K\in \Bbb{R}^n$ is compact. Let us denote $D=\sup_{x,y\in K}|x-y|$ as $K's$ diameter. Prove there exist $a,b\in K$ such that $D=|a-b|$ i.e, that the suprimum is the maximum. I know there ...
-1
votes
1answer
39 views

Mathematical pre-requisites to read History of Topology by I. M. James [on hold]

What are the necessary mathematical pre-requisites to read History of Topology by I. M. James?
1
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0answers
30 views

A set $A\in \Bbb{R}^n$ is compact $\iff$ every continuous function on $A$ is bounded. [duplicate]

A set $A\in \Bbb{R}^n$ is compact $\iff$ every continuous function on $A$ is bounded. I have a problem proving the direction according to which $A$ is compact. First direction I said: If $A$ is ...
1
vote
1answer
41 views

Lexicographic Orderings of Aronszajn Trees are Aronszajn Lines

I am trying to prove that every lexicographic ordering of a Aronszajn tree is a Aronszajn Line. If $T$ is a tree, a lexicographic ordering of $T$ is defined as follows: For each ...
4
votes
2answers
46 views

(A question regarding:) the graph associated with an open cover of a topological space.

Let $X$ denote a topological space and suppose that $\mathcal{O}$ is an open cover of $X$. Assume $\emptyset \notin \mathcal{O}.$ (Thanks Niels!) Now make $\mathcal{O}$ into an (undirected) graph as ...
3
votes
1answer
65 views

Density of sin(k) where k is an integer [duplicate]

Consider the set $A=\{\sin k:k\in\mathbb Z \}$. I want to know whether this set is dense in $[-1,1]$. I have a hunch that this problem can somehow be reduced to the approximation of $\pi$ using ...
2
votes
0answers
40 views

Connected set imply continuous boundary

Is it true that a connected and bounded set of $\mathbb{R}^2$ has a boundary that can be parametrized by a continuous mapping?