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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56 views

Density and convergence

I have a small question: Is it true that if the basis of a space $A$ is dense in a space $B$ ($B\subset A$) then if $u_n\rightarrow u$ in $A$ we have that $u_n\rightarrow u$ in $B$ ?
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1answer
58 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
51 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
77 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
69 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
15 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
33 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
16 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
52 views

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

Why $ \Omega \neq \bar{\Omega} \setminus \partial \Omega $ ? Can somebody show any counterexample?
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0answers
22 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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0answers
25 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 ...
3
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1answer
30 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
44 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
46 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 ...
2
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1answer
61 views

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

Let $A$ be a compact connected subset of the plane. Is $\Bbb R^2\setminus A$ connected? Why?
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1answer
34 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
32 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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0answers
47 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 $$ \quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}} $$ ...
2
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1answer
98 views

The Zariski topology on $\operatorname{Spec} A$ as an intial topology

Given any commutative ring $A$ let $\operatorname{Spec} A$ 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 ...
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1answer
49 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
39 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
61 views

Is $\sin (\mathbb N)$ dense in $[-1,1]$? [duplicate]

Let $\mathbb N$ be the set of positive integers, then is it true that $\sin (\mathbb N)$ is dense in $[-1,1]$ i.e. is it true that for every $x,y \in [-1,1]$ with $x<y$ , $\exists m \in \mathbb N$ ...
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0answers
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
67 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
50 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 ...
2
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2answers
28 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
41 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 ...
2
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1answer
31 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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0answers
70 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
77 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 ...
2
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1answer
63 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 ...
9
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1answer
84 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
45 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
128 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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0answers
55 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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0answers
26 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: ...
2
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1answer
28 views

the homeomorphisms betwen two spaces looks like broom

Let $Y=${$(x,x/n)\in \mathbb{R} \times \mathbb{R}: x\in [0,1],n \in \mathbb{N}$} and $X=\cup_{n\in \mathbb{N}}{[0,1]\times (n)}$ and $(0,n) R (0,m),\forall n,m \in \mathbb{N}$. Then does $X/R $ is ...
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2answers
54 views

Sequentially compact space

Is every sequentially compact space metrisable? If not, then, can you give me an example of a sequentially compact space that is not compact.
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2answers
38 views

Is this a quotient map?

Consider the map $f \mapsto f(0)$ from $\mathcal C([0,1])$ into $\mathbb R.$ Here $\mathcal C([0,1])$ is the space of continuous real functions on $[0,1]$ with the usual sup metric. Show that this is ...
2
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1answer
54 views

can a compact set have infinite measure?

Can a compact set have infinite measure? It does not seem to violate the measure axioms. This is not true in the case of Lebesgue measure. So I am also wondering is there any clean cut condition for ...
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2answers
39 views

Is Lower limit topology $\mathbb{R}_l$ finer than the standard topology $\mathbb{R}$?

Is Lower limit topology $\mathbb{R}_l$ is finer than the standard topology $\mathbb{R}$? In Munkres' topology, it's stated that $\mathbb{R}_l$ is finer than $\mathbb{R}$. In the argument , he is ...
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1answer
66 views

Locally compact groups

Let $S= G_1\bigcup G_2 $, where $G_1$ and $G_2$ are two groups. If $S$ is locally compact, is it true that either $G_1$ or $G_2$ is locally compact? Thank you.
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3answers
80 views

Suggestions for a real analysis reference.

Can anyone suggest some real analysis book which has a geometric presentation of the concepts with pictorial representation.
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1answer
44 views

Are circulant matrices open

Are the set of positive definite symmetric circulant matrices open in the set of positive definite symmetric matrices?
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0answers
37 views

explicit function between transformation matrix and vertex in polyhedron

recently I am stuck in solving a geometric problem. I hope someone could give me some tips, thanks for all in advance!!! Question 1: given a constant polygon $M1$ with 4 vertices: ...
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2answers
51 views

Sequence of compact sets

Let $(X,d)$ be a metric space and consider an increasing sequence $A_n$ of its subsets such that $A = \bigcup_n A_n$ is compact. Can it happen that $A\setminus A_n$ is compact for all finite $n$?
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1answer
32 views

Proof of an open set or closed set

I'm struggling on a proof that I can't proof correctly. Let $A=\mathbb{Z}$, $B=\{n-\frac{1}{2n} | n \in \mathbb{N}*\}$ I could prove easily that A is a closed set and B as well : $\overline A =$ ...
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0answers
25 views

What does it mean to say “Resolving intersections”

Consider a surface (with boundary) $S$ with marked points on the boundary such that we may may triangulate the surface. Call a line joining two marked points in a triangulation an arc. Consider a ...