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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Proper maps and Hausdorff Space

Now, I need help with this exercise: Let $(X,\tau)$ and $(Y,\sigma)$ two topological spaces. Let $f:X\to Y$ a continous map such that the preimage of a compact subset in $Y$ is compact in $X$. Show ...
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3 views

Interior and closure of a Dirichlet domain in a Riemannian manifold

Let $X\neq\varnothing$ be a complete connected Riemannian manifold. Suppose $G$ is a group of isometries of $X$, acting properly discontinuously on $X$. We assume there is a point $x_0\in X$ such that ...
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1answer
63 views

If $f\tau$ is continuous for every path $\tau$ in $X$, is $f:X\rightarrow Y$ continuous?

Let $X$ be a path connected space and $Y$ be a topological space. Let $f:X\rightarrow Y$ be a function such that for every path $\tau:\mathbb{I}\rightarrow X$ , $f\tau:\mathbb{I}\rightarrow Y$ is ...
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1answer
27 views

Are Hausdorff compactifications of a Tychonoff space $X$ in one-to-one correspondence with completely regular subalgebras of $BC(X)$?

Let $X$ be a completely regular (Tychonoff) topological space. It is known that if $\mathscr F\subseteq C(X,[0,1])$ separates points and closed sets (that is, for every closed set $E\subseteq X$ and ...
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70 views

What are the properties of the set of the Real Numbers without the Integers?

This question came up in a lunchtime discussion with coworkers. None of us are professional mathematicians or teachers of math. I apologize for any incorrect math or sloppy terminology. We were ...
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1answer
17 views

Interior, closure, boundary of countable complement topology

I've been set the following question I started by trying to work out the interior, but I couldn't get any further. I'm using the definition that the interior of a set $A$ w.r.t. a topological ...
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24 views

Separable metrics

Is the following true?: Let $d_1$ and $d_2$ two separable metrics in space $X$. Then $d=\max(d_1,d_2)$ is a separable metric on $X$. Thanks!
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9 views

How to generate a Poincare section for discrete particle trajectory?

I'm a novice when it comes to generating Poincare sections, and I can't seem to get it right. I have a particle moving in a 3D periodic field, and I wish to generate a Poincare section of its ...
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2answers
29 views

How does reparametrization change the shape of a curve?

This is a question about the reparametrization of a curve. Say $\psi:[0,1]\to [0,1]$ is a continuous mapping where $\psi(0)=0$ and $\psi(1)=1$. Also, $f:[0,1]\to X$ is a continuous mapping; a ...
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27 views

Problem about base (topology)

Consider following 3 subsets of $\mathbb R^2$. Let $Σ^2$ consists of all possible open disks, $Σ^∞$ consists of all possible open square with sides parallel to the coordinate axis, and $Σ^1$ ...
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1answer
27 views

Covering $S^{n-1}$ with $n+1$ closed sets containing no antipodal points

For proving the equivalence of the Theorems of Borsuk-Ulam and Lusternik-Schnirelmann, we need to cover $S^{n-1}$ with closed sets $F_1,\dots,F_{n+1}$ such that none of the $F_i$ contains a pair of ...
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12 views

A Lindelof non-scattered space $X$ which is not an extention of $\mathbb R$

Is anyone familier with an example for a Lindelof non-scattered topological space $X$ which is not an extention of $\mathbb R$ (with Euclidean topology). I am looking for an example which is not a ...
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1answer
46 views

An ultrafilter product topology

Suppose $X=\prod _{i\in\omega}X_i$ is the cartesian product of topological spaces $X_i$ and $u$ is a filter on $\omega$. Define a basis for $X$ by taking the collection of all sets of the form ...
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1answer
50 views

Is an algebra the smallest one generated by a certain subset of it?

Let $X$ be a completely regular topological space and let $BC(X)$ denote the space of bounded continuous complex-valued functions on it. Also, let $C(X,[0,1])$ be the set of continuous functions on ...
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1answer
17 views

Open ball with rational radii forms a basis.

Show that in a metric space, the set of all balls with rational radii is a basis for the topology. Although I understand the question, I have no clue at all. I am very new to topology, can anyone ...
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1answer
48 views

continuous and COVERING MAP

Let $p:E\rightarrow B$ be a covering map. Let $Y$ be locally path-connected. Let $g:Y\rightarrow E$ be a function ( which we do not assume is continuous) such that $p\circ g$ is continuous, and ...
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2answers
63 views

The closure and the boundary of $\mathbb{R}^{\infty}$ in $\mathbb{R}^{\mathbb{N}}$.

I think that in: Product topology: $\overline{\mathbb{R}^{\infty}}=\mathbb{R}^{\mathbb{N}}$ and $\partial\mathbb{R}^{\infty}=\mathbb{R}^{\mathbb{N}}$. Box topology: ...
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1answer
14 views

Dense subset relation

Defn Let $B$ be a Boolean algebra. A subset $D$ of $B$ is called b-dense if for every $0\neq b\in B$, there is $0\neq d\in D$ such that $d\leq b$. Defn Let $T$ be a topological space. A subset $D$ of ...
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1answer
31 views

Attaching space, help on visualization

Let $X$ be a topological space, $A\subset X$ a closed subspace. $CA$ means the cone of $A$, and by $SA$ I'll denote the suspension of $A$. I need to prove that $$ \left( \left( (X \cup CA) \cup ...
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0answers
14 views

Proof verification related to the discrete metric

Can someone please verify my proof? Let $X_1$ be a set and let $d_1$ be the discrete metric on $X_1$. (a) Prove that every subset of $(X_1, d_1)$ is open. (b) Prove that if $(X_2, d_2)$ ...
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1answer
49 views

Why set is not equal its closure minus its boundary? [on hold]

Why $ \Omega \neq \bar{\Omega} \setminus \partial \Omega $ ? Can somebody show any counterexample?
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0answers
20 views

Prove that $d_\infty(f, g) = \operatorname{sup}\{|f(x)-g(x)|:x \in [a,b]\}$ defines a metric

Can someone please verify my proof? Let $C[a,b]$ denote the set of all continuous functions from $[a,b]$ to $\mathbb{R}$. Let $d_\infty:C[a,b] \times C[a,b] \longrightarrow [0, \infty)$ be given ...
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31 views

Comapctness property [on hold]

Is the arbitrary union of compact sets is compact in $R^2$? The finite union of compact sets is compact but what about infinite union i don't know.If it is not compact then please give one counter ...
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21 views

Almost Everywhere Function Space

Problem Let $\Omega$ be a measure space with measure $\mu$ and $V$ a topological vector space not necessarily Hausdorff as well as the function space $\mathcal{F}:=\{f:\Omega\to V\}$ topologized by ...
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1answer
29 views

Equivalence of relative and (reduced) homology for arbitrary pairs

I could not find my mistake in the following argument, though I know it is wrong. This is more like a "Q&A", since there is nothing to "prove" in the positive sense. Here it goes: For an ...
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1answer
43 views

Continuous function between topological spaces

Let $ (X,\tau_{X}) $ and $ (Y,\tau_{Y}) $ be topological spaces and $ f:X\rightarrow Y $ be a function. My question is how to show if for each $ A\subseteq Y $ , $\overline{f^{-1}(A)}$ $ \subseteq ...
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1answer
29 views

Topology Book including specific aspects

I am looking for a basic book about Topology (maybe also a bit of Functional analysis but basically Topology) including the following points (in addition to the basic points): $\bullet$ Seminorms ...
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1answer
59 views

If $A$ is compact and connected, then is $\Bbb R^2\setminus A$ connected? [on hold]

Let $A$ be a compact connected subset of the plane. Is $\Bbb R^2\setminus A$ connected? Why?
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1answer
33 views

Hausdorff or weaklly hausdorff may apply

Let $X$ be a topological space and suppose that there is a countable collection of open sets $$\mathbb{B}\{U_1,U_2,…\}$$ which is a basis for the topology of $X$. Let $A\subset X$ and let $x\in ...
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2answers
28 views

Existence of a boundary point

I am not particularly well-versed in topology, so I wanted to check with you whether there exists a much simpler argument to prove the following statement or whether there are problems with my proof. ...
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40 views

$ d(x,y)^2 \le d(x,z)^2 - d(y,z)^2\color{}{+\varphi\big(x,y,z,d(x,y),d(y,z) \big)}? $

In a metric space $(M,d)$ the triangle inequality $d (x, z) \le d(x, y) + d (y, z)$ gives us's the inequalitie $$ \quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}} $$ ...
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1answer
86 views

The Zariski topology on $\mathrm{spec} \ A$ as an intial topology

Given any ring $A$ let $\mathrm{spec} \ A$ be the space of prime ideals of $A$. Can we interpret the Zariski topology as an initial (or final) topology with respect to some canonical maps from ...
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1answer
46 views

Homeomorphism are equivalence relations, so what are the equivalence classes?

Homeomorphisms are equivalence relations, so what are the equivalence classes for two Topological spaces $T_1, T_2$? Intuitively it seems like we might have the following equivalence classes - ...
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1answer
37 views

Intuition behind homeomorphism from $B((0, 0), 1) \to \mathbb{R^2}$

In my notes I have that the following function is a homemorphism from $B((0, 0), 1) \to \mathbb{R^2}$ $$h(x, y) \to \frac{f(\sqrt{x^2 + y^2})}{\sqrt{x^2 + y^2}} (x, y)$$ where $f = ...
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0answers
25 views

First countable space with a sequence in it having a cluster point [on hold]

Let $( x_n )$ be a sequence in a topological space $X$ ,having a cluster point $x \in X$. Prove that if $X$ is first countable at $x$ then there exists a subsequence of ${ x_n}$, which converges to ...
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19 views

Showing that bd$A$=$\{\vec v\in\mathbb{R}^n| d(\vec u,\vec v)<r\}$

Problem: And $\beta_r(\vec u)\equiv \{\vec x\in\mathbb{R}^n| dist(\vec u,\vec x)<r\}$. I got the first part showint that Int $A$=$A$. Now I want to show that bd$A$=$\{\vec v\in\mathbb{R}^n| ...
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1answer
64 views

Continuity of distance function without triangle inequality

We say that a continuous function $\rho : \mathbb{C}^n \to \mathbb{R}$ is a distance function if the following three conditions hold: 1.) $\rho \geq 0$ 2.) $\rho (z) =0$ iff $z=0$ 3.) $\rho(cz)= ...
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2answers
48 views

Can any collection of open sets in $\mathbb{R}$ be covered by a countable subcollection?

Let $A$ be a collection of open sets in $\mathbb{R}$. is there a countable subcollection $G_i$ of $A$ such that $$\cup_{G\in A} G=\cup_{i=1}^\infty G_i$$ I guess there must be such subcollection, but ...
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2answers
26 views

Doubt about limit point

I came though this definition about limit point A point z is a limit point for a set A if every open set U containing z intersects A in a point other than z. I want to know can we change it to ...
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1answer
34 views

Fibrewise product

I have recently started studying fibrewise topology. It is not clear to me what is the difference between the normal product space and the fibrewise product space over a topological space B. I am ...
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1answer
27 views

Continous surjective map from $S^1$ to $S^n$

Is there any continous surjective map from $S^1$ or $[0,1]$ onto $S^n$, for some $n\geq 2$. Thank you.
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68 views

Closed and Connected subgroups of $\mathbb{R}^n$

Question is : What are closed connected subgroups of $\mathbb{R}$ and from that deduce what are closed connected subgroups of $\mathbb{R}^n$ What i have done so far is : Only connected subsets of ...
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2answers
74 views

Formally show that the set of continuous functions is not measurable

Let $C(\mathbb{R})=\{ f:\mathbb{R}\to \mathbb{R} \colon \ f \text{ continuous}\}\subseteq \mathbb{R}^{\mathbb{R}} $. How to prove formally that $C(\mathbb{R}) \notin ...
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1answer
57 views

Is $[0,M]^\infty $ connected and separable space?

I know that $[0,M]\subset R_+ $ is connected, separable. Now, let us consider the infinite dimensional space $[0,M]^\infty $. I want to see whether this space in connected and separable. I think the ...
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59 views

Packing infinitely many ellipses into a circle

Given a circle $C$, and an infinite set $S$ of mutually disjoint ellipses which are inside and tangent to $C$, prove that there must exist a disk $D$ which lies inside $C$ but outside every ellipse. ...
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1answer
44 views

Orthogonality on Banach spaces

I got a doubt with a proof in Brezis' Functional Analysis, theorem 2.16. It says Theorem 2.16: Let $G,L \subset E$ be two closed subspaces in a Banach space $E$. Then the following properties are ...
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4answers
113 views

$[0, 1)$ and $S^1$ not homeomorphic?

Let $f:[0, 2\pi) \to S^1 = \{(x, y): x^2 + y^2 = 1\}$ be such that $f(t) \to (\cos t, \sin t)$ $f$ is a continuous bijection but it is NOT a homeomorphism. I suppose the only point of contention is ...
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1answer
41 views

Continuous function - unsure of statement that lacks rigour

I have the following statement in my Topology notes in a section on continuous functions - A polynomial of degree $n$ has at most $n$ roots. Thus $f^{-1}(b)$ is finite. This shows that ...
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23 views

The future of the orbit of a point is a closed set [duplicate]

$X$ is a metric space and $f: X \rightarrow X$ is a dynamical system. Prove: $w(x_{0})$ is closed. Here the set $w(x_{0})$ is the future of the orbit of $x_0$, defined as $$\omega(x_0) = \{y \mid ...
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1answer
26 views

Any polynomial function is continuous - what about a constant function?

I read that any polynomial function is continuous. I.e. If we have an open set $U$ in the range, $f^{-1}(U)$ will be open in the domain. Let $\mathbb{R}$ have the standard topology. Define $f: ...