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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77 views

Cartesian product of KC spaces

We know that Cartesian product of KC spaces do not need to be a KC space. Is it true to say" if $X$ is a $KC$-space then for each $k ‎\geq2$ ,$X^k$ is $KC$ iff each compact subspace of $X$ is ...
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0answers
66 views

What is the relation between singular point for a function and the one in a vector field?

What is the difference between sigular point for a function and the one in a vector field? Is the derivative or divergence at the singular point must be infinity? By the way, what is the relation ...
2
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2answers
951 views

Is this proof that all metric spaces are Hausdorff spaces correct?

Let $x$ and $y$ be distinct points of a metric space $M$. Prove that there exist in $M$ disjoint open sets $U$ and $V$ with $x \in U$ and $y \in V$. Let $U$ and $V$ be open balls centered at $a$ and ...
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2answers
144 views

Algebraic Topology: CW complexes and their associated $n$-skeletons questions.

The following is from Hatcher's book on Algebraic Topology: we first start with a discrete set $X^0$ whose points are $0$-cells. Then "Inductively, form the n skeleton $X^n$ from $X^{n−1}$ by ...
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1answer
107 views

Etale spaces of a presfeaf and the associated sheaf

Given a presheaf $\mathcal{F}$on a topological space $X$, one can construct the etale space $\pi_1 : Y_1\to X$. Let us now look at the associated sheaf $\mathcal{F}^+$ as a presheaf and construct the ...
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2answers
264 views

Lebesgue measure paradox

See Lebesgue outer measure of $[0,1]\cap\mathbb{Q}$ Lebesgue measure: $$ m(A) = \inf \left\{ \sum |I_n| : A \subset \bigcup I_n \right\} $$ We know that $m(\mathbb{Q} \cap [0,1]) = 0$. Proof: ...
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1answer
74 views

Is every continuous closed surjection also open?

$f:X\rightarrow Y$ is a continuous closed surjection, $X$ and $Y$ are topological spaces. Is $f$ also open?
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3answers
415 views

Path Connectedness of a Set

I am finding it very difficult to prove or disprove the following statement. If $A$ is a family of countably many lines in $\mathbb{R}^3$ then $\mathbb{R}^3\setminus A$ is path connected. I would ...
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3answers
324 views

Metric spaces and openness

I am asked to prove two things. I would like to know if the proof was elaborate and concise. I would also like to know if proving reductio ad absurdum is looked down upon. I have heard from my ...
4
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4answers
381 views

Why sets that aren't closed can't be compact?

In $\mathbb{R}^n$ we prove that a set is compact (using the definition about open covers) if and only if it's closed and bounded. It is pretty clear that if $\mathcal{O}$ is an open cover of one ...
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1answer
287 views

Rudin's PMA Exercise 2.18 - Perfect Sets [duplicate]

I've been working through Chapter 2 questions and have thought about Exercise 2.18 for a while, but couldn't come up with an answer. Is there a nonempty perfect set in R which contains no rational ...
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2answers
136 views

Define a metric based on a topology

Is their a systematic way, based on a topology meterizable $(X, t)$ to define or compute some metric $d$ on $X$ such that the open balls in $(X, d)$ is a metric of $(X, t)$?
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1answer
262 views

Connectedness in Matrices

Are orthogonal matrices connected?? What about Unitary matrices and Normal matrices?? I would also like to know about the path connectedness in them(Need an explanation).
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1answer
41 views

How to decide if these two maps are proper?

We define a mapping $f$ of a topological space $X$ into a topological space $Y$ to be proper if the subspace $f^{-1}(C)$ is compact in $X$ whenever $C$ is a compact subspace of $Y$. Now how to ...
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1answer
101 views

The boundary of a unbounded simply connected planar domain

Let $D$ be a unbounded simply connected planar open domain. Let $\partial D$ be its boundary. The question is the following: Can $\partial D$ have a compact component?
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1answer
203 views

Archimedean property application

The Archimedean property states that for all $a \in \Bbb R$ and for some $n \in \Bbb N: a < n$. Similarly, for all $n \in \Bbb N | b \in \Bbb R | 0 < {1 \over n} < b$. Thus, there ...
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1answer
164 views

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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0answers
61 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
136 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 ...
2
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0answers
76 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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3answers
113 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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1answer
51 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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3answers
2k views

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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1answer
56 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
146 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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1answer
45 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 ...
2
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2answers
112 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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1answer
179 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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2answers
234 views

T4 and first countable topology that is non metrizable

Does anyone know any example of such topology?
12
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2answers
216 views

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

It is well known that any continuous map between 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 ...
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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
51 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
49 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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3answers
484 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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1answer
68 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 ...
2
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1answer
93 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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2answers
99 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 ...
3
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1answer
123 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
41 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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2answers
102 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
98 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
84 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
139 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
192 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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50 views

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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1answer
80 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 ...
2
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1answer
112 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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3answers
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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3answers
347 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 ...