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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If we have an embedding $f:X \rightarrow A$, where $A \subset Y$, do we have to show $f^{-1}$ is continuous?

If we have an embedding $f:X \rightarrow A$, where $A \subset Y$, do we have to show $f^{-1}$ is continuous? I'm looking at a proof where they only show that $f$ is continuous and 1-1. Then I looked ...
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63 views

Error in Lang's definition of weak topology?

On page 23-24 of his Real and Functional Analysis (3e) Serge Lang claims Let $Y$ be a topological space and let $\mathscr{F}$ be a family of mappings $f \colon X \to Y$ of $X$ into $Y$. Let ...
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1answer
145 views

Continuous function and normal topological space

Let $X$ be a normal topological space. If $A \subset X$ is closed and $G_{\delta}$, then there exists a continuous function $f:X \to [0,1]$ such that $f(x) =0$ if $x \in A$ and $f(x) \neq 0$ if $x ...
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83 views

Is there a $P$-space linearly Lindelöf and non-Lindelöf?

A completely regular topological space $(X,\tau)$ is a $P$-space, if every $G_\delta$-subset of $X$ is open (i.e $\tau$ is closed under countable intersection). A topological space $X$ is linearly ...
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115 views

compactness of topological space

i would like to understand easily notation of compact space,i had read that space is compact if it is closed and bounded,fr example following link says that The closed unit interval $[0,1]$ is ...
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54 views

definition of accumulation point

is correct the following definition? -- "let $ s \in \mathbb{R} $ and $ T \subseteq \mathbb{R} $, $ s $ is accumulation point for $ T $ if $ \forall S \in \mathcal{U}(s)((S-\{s\})\cap T \neq ...
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How can the set $\{1\}$ be in the co-finite topology?

I would like to clarify the definition of the co-finite topology. The general definition says this: Let $X$ be a non empty set. Then the collection of subsets of $X$ whose compliments are finite ...
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57 views

Contractibility of the total space of the infinite tautological bundle minus the zero section

There is a tautological line bundle $L$ on the infinite dimensional projective space $\mathbb{RP}^\infty$ with total space $L=\{(x,y)\in \mathbb{RP}^\infty\times \mathbb{R}^\infty\mid v\in x\}$ and ...
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1answer
164 views

Fuchsian groups and topological isomorphism

I have a (finite) presentation of a group and I am wanting to prove that it is not Fuchsian. Because it is given by a presentation, a neat, algebraic description of Fuschian groups would be nice. This ...
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47 views

Structures on spaces of topologies

Does there exist a fruitful notion of "moduli space of topologies"? For example, is it possible to define useful/natural topologies on the set of topologies on a given set $A$? When does it make ...
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2answers
127 views

Homeomorphism is to topology as continuity is to

I have two questions. Are there examples where continuity does not preserve connectedness? Is there a structure whose structural properties are preserved by continuous map? (Just like homeomorphism ...
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192 views

example of quotient topology

if we work in $\mathbb{R}^2\(0,0)$ with euclidean topology and we set following equivalence relation $P$ on this space: $(x,y)P(x',y')$ iff there exists $a$ in $\mathbb{R}^2\(0,0)$ such that $(x,y) ...
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257 views

T4 and first countable topology that is non metrizable

Does anyone know any example of such topology?
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249 views

Are compact spaces characterized by “closed maps to Hausdorff spaces”?

It is well known that any continuous map from a compact space to a Hausdorff space must be a closed map. Does this fact characterize compactness? That is, if for a space $X$, every continuous map to ...
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0answers
44 views

Is this proof correct: domain of a quotient map $p:X\rightarrow Y$ is connected when $p^{-1}(y)$ is connected for each $y\in Y$ and $Y$ is connected.

The domain $X$ of a quotient map $p:X\rightarrow Y$ is connected when $p^{-1}(y)$ is connected for each $y\in Y$ and $Y$ is connected. Proof: If $X = F \uplus G$ for two nonempty closed sets $F,G$ ...
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1answer
54 views

Connectedness of sets acting on topological groups…

I come now with a topological group question. Suppose a topological group $G$ acts on a topological space $X$. Suppose $G$ and $X/G$ are connected. Show $X$ is connected. Me and a few friends have ...
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0answers
52 views

JSJ-decomposition of a non-hyperbolic 3-manifold

Suppose $M$ is a $3$-manifold. Then you can split it over spheres. This is the "prime decomposition" and is unique. You can then split the components of this decomposition along tori. If you leave the ...
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587 views

Why do we need Hausdorff-ness in definition of topological manifold?

Suppose $M^n$ is a topological manifold, then $M^n$ locally looks like $\mathbb{R}^n$. $M^n$ is locally Hausdorff, since $\mathbb{R}^n$ is Hausdorff and Hausdorff-ness is a topological invariant. ...
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76 views

$X$ a separable metric space with no isolated points. If $G \subset X$ is a countable dense $G_{\delta}$ subset of $X$, why is $X$ meager?

Let $X$ be a separable metric space with no isolated points. Then if $G \subset X$ is a countable dense $G_{\delta}$ subset of $X$, why is $X$ meager? I've written a couple things so far, Let ...
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1answer
95 views

countably compact, KC minimal

** Lemma :Let $(X,\tau )$ be a KC-space which is not countably compact. Then X can be condensed onto a weaker KC-topology.** Proof: Let new topology $‎ ‎\tau‎^{‎\prime‎} = ‎\{U‎‎\in‎‎ ‎\tau:‎‎ ...
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103 views

Monotone convergence example

In the first chapter of Probability wih Martingales (Willams) I came across the following example. Book says it's wrong, I don't understand what is wrong in that. Could somebody please explain why ...
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1answer
131 views

Which of the given statements are true?

Which of the following statements are true? a. Consider the subspace $S^1 = \{(x, y)\in \mathbb{R}^2:x^2 + y^2 = 1\}$ of $\mathbb{R}^2$. Then, there exists a continuous function $f : S^1\to ...
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1answer
42 views

Gillman & Jerison on chains of ideals in $C(N)$

I'm looking at problem $\mathbf{2J}$ in Gillman & Jerison's Rings of Continuous Functions. Specifically, parts 2 and 3 as follows. (Relevant definitions are at the bottom.) 2. Find a chain of ...
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106 views

Intrinsic thought of relative topology

I am learning topology and I found some difficulties in relative topology of $\mathbb R^2$ usual space. For example, let $Y=\{x:d(x,<0,0>)=1\}\setminus\{<1,0>\}$ where $d$ is the ...
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1answer
102 views

Pick out the true statements.

Pick out the true statements. a. Let $f : \mathbb Z\to \mathbb Z^2$ be a bijection. There exists a continuous function from $\mathbb R$ to $\mathbb R^2$ which extends $f.$ b. Let $D$ ...
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1answer
94 views

Question about a base for a topology

Actually, this is only a clarification about the definition of a base for a topology. In the book of Dshalalow entitled "Real Analsysis: An Introduction to the Theory of Real Functions and ...
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1answer
44 views

Prove that $Ω$ is closed in the standard topology

The motivation to this question can be found in: Prove that $Ω$ has no accumulation point My question is: Prove that $Ω$ is closed in the standard topology.
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1answer
141 views

Typo or additional hypotheses needed in problem 5, p.71 of Bredon's Topology and Geometry?

The problen can be foun in p.71 of Topology and Geometry. I state it below for convenience. Problem: Consider the half open real line $X=[0,\infty)$. Define a functional structure $F_{1}$ by taking ...
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1answer
203 views

Are $\mathbb{Q}_p$ and $\mathbb{Q}_q$ homeomorphic?

If $p$ and $q$ are distinct prime number, are $\mathbb{Q}_p$ and $\mathbb{Q}_q$ homeomorphic as topological space?
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Is it possible to say that the set $D$ is discrete?

Assume that a set $D$ has no accumulation point. Then: Is it possible to say that the set $D$ is discrete?
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86 views

Is a weakly contractible connected metric space, uniquely geodesic?

A topological space is weakly contractible if all the homotopy groups are trivial. It's connected if it's not the union of two disjoint nonempty open sets. A metric space $(X,d)$ is uniquely geodesic ...
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1answer
116 views

Prove that $D$ is a simply connected domain

Definition. A region $D$ is said to be simply connected if any simple closed curve which lies entirely in $D$ can be pulled to a single point in $D$ (a curve is called simple if it has no self ...
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63 views

The set $A=\{(x, y, z)\in\mathbb{R}^3:x^2+y^2\leq 4\}$ is closed

I need to prove that the set $A=\{(x, y, z)\in\mathbb{R}^3:x^2+y^2\leq 4\}$ is closed, to do this give me any point $p=(a, b, c)\in A^c$ and define $$r=\sqrt{a^2+b^2}-2=||(a, b, c)-(0, 0, c)||-2$$ I ...
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380 views

How does definition of nowhere dense imply not dense in any subset?

In some topological space $X$, a set $N$ is nowhere dense iff $\text{Int}\left(\overline{N}\right)=\emptyset$, where Int is the interior, and overbar is closure. How can I show this is equivalent to ...
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1answer
304 views

How to show that a disjoint sum of metric spaces is metrizable?

I have encountered this question while tring to figure out why every sequential space $X$, is a quotient spaces of a metric space. If I understand correctly, given a sequential space, every sequence ...
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1answer
48 views

Is the image of a continuous idempotent necessarily homotopic to the original space?

Let $f$ be a continuous self-map of a topological space $X$ such that $f\circ f=f$. Is it true that $X$ is homotopic to its image $f(X)$?
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646 views

Sphere-sphere intersection is not a surface

In my topology lecture, my lecturer said that when two spheres intersect each other, the intersecting region is not a surface. Well, my own understanding is that the intersecting region should look ...
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1answer
88 views

Why does a subcomplex of a cell complex being closed mean the characteristic map has an image in the subset?

I was trying to learn some Algebraic Topology though I haven't got very far yet so I would greatly appreciate it if you gave as simple answer as possible. On page 7 of Hatcher he says: Since $A$ ...
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1answer
115 views

Help with a proof of a theorem about convex sets

I'm studying the following theorem: Theorem 1. Let $C$ be a convex set and let $\textbf{y}$ be a point exterior to the closure of $C$. Then there is a vector $\textbf{a}$ such that ...
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186 views

Is $\{x\}$ a neighbourhood of $x$?

According to Wikipedia: a neighbourhood of a point $x\in S$ is a set $U=\{y\in S|\exists\varepsilon>0:\|x-y\|<\varepsilon\}$. My question is: is $\{x\}$ a neighbouthood of $x$ (probably not, ...
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0answers
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Cantor set homeomorphism [duplicate]

I am trying to prove that $\{0,1\}^\mathbb{N}$ is homeomorphic to the cantor set. Consider the mapping $f:\{0,1\}^\mathbb{N}\to[0,1]$ defined as$$f(x)=2\sum_{n=0}^\infty 3^{-n-1}x(n)$$ I think that ...
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99 views

Prove that $Ω$ has no accumulation point

Let $f,g,h,l:ℂ→ℝ$ four harmonic functions such that $f≠g$ and $h≠l$. Let $D$ be an open set in $ℂ$. Let us define the set: $$Ω:\{ s=α+iβ∈D:f(s)=g(s),h(s)=l(s)\}$$ My question is: Prove that $Ω$ ...
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92 views

Intrersection of open set is closed

In the first chapter of Probability wih Martingales (Willams) I came across the following relation $$(-\infty,x]=\bigcap_{n\in N}(-\infty,\, x+n^{-1}).$$ Even though this is intuitive to me I would ...
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Does an equivariant weak equivalence induce weak equivalences on all orbits?

This question arose from another, which was not well formulated and completely answered by this MO thread as pointed out by the user roman. Let $G$ be a discrete group, $X$ and $Y$ be $G$-spaces (and ...
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1answer
200 views

$f$ holomorphic from unit disc to itself. $f\left(\frac{1}{2}\right) = f\left(-\frac{1}{2}\right) = 0$. Show that $|f(0)| \le 1/3$.

I'm studying for a qual exam. I cannot solve the following problem: Let $f$ be holomorphic from the unit disc to itself. $f\left(\frac{1}{2}\right) = f\left(-\frac{1}{2}\right) = 0$. Show that ...
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A “non-trivial” example of a Cauchy sequence that does not converge?

A Cauchy sequence doesn't necessarily converge, e.g. take the sequence $(1/n)$ in the space $(0,1)$. Maybe my intuition is wrong but I tend to think of this as, "it does converge but what it ...
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79 views

ultrafilter $\mathcal{F}$

I have a question about relation between member of a ultrafilter $\mathcal{F}$ on a topological space $(X,\tau)$ and a subset $K$ of $X$. Can we cosider a uniform ultrafilter $\mathcal{F}$ as a ...
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70 views

When is a $C^1$-function $f:\mathbb R^m \to \mathbb R^n$ Lipschitz-continuous?

Let $f:\mathbb R^m \to \mathbb R^n$ be continuously differentiable. I'm looking for a proof that there is an $M>0$ such that $\|f(x)-f(y)\| \le M \| x-y\|$ for all $x,y \in A=[-1,1]^m \subset ...
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322 views

$\mathbb S^3=\mathbb R\mathrm P^3$? Find mistake in the proof.

It is known that the sphere $\mathbb S^3$ may be decomposed as two solid tori. It may be done with the following "parametrization" $f:(\mathbb S^1\times \mathbb S^1 \times [0,1])_{/\sim}\to \mathbb ...
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1answer
247 views

A functional structure on the graph of the absolute value function

Let $X$ be the subspace of $\mathbb{R}^2$ consisting of the graph of the absolute value function. That is, $X=\{(x,|x|) : x\in\mathbb{R})\}$. We define a functional structure on $X$ by restricting ...