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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7 views

Number of cells in a minimal cell structure for a non-simply connected manifold?

I have obtained a cell structure of a connected (but not simply connected) manifold using Morse theory. Is there any way for me to know whether this cell structure is minimal?
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1answer
9 views

Can we construct from $[0,\omega_1)$ a space which is strictly-Frechet with no winning strategy in $G_{np}(q,E)$?

I have asked in here a question which tured out to make no sense. I think I have found the confusion and would like to try and rephrase my question: Let $E$ be a topological space, $q \in E$. ...
-1
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1answer
34 views

Topologies homeomorphism [on hold]

Let $(X,\tau)$ be a topological space, where $X$ is an infinite set. Show that $(X,\tau)$ has a subspace homeomorphic to $(\mathbb{N},\tau_{1})$, where either $\tau_{1}=\{\mathbb{N},\emptyset\}$, or ...
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2answers
114 views

Homeomorphism group

I need to solve this problem, but I don't know how start. Let be $(X,\tau)$ a topological space, and $G$ the set of all homeomorphism of $X$. I just proved that $G$ is a group, but I need also prove ...
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1answer
36 views

How do i show that if every continuous function on $X$ is bounded, then $X$ is compact? [duplicate]

Let $(X,d)$ be a metric space. Assume every continuous function on $X$ is bounded. Prove that $X$ is compact. Well, i don't know which continuous function should i fix to start an ...
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1answer
38 views

How do i prove this is a metric?

Define a metric $d$ on $\mathbb{Z}$ in the following manner: $d(x,y)=\min\{\frac{1}{n!} : n! \text{ divides } x-y \text{ where } n\in\mathbb{Z}^+ \}$ if $x\neq y$. $d(x,x)=0$ ...
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0answers
40 views

how do i prove that the the set of irrationals cannot be a countable union of closed subsets? [duplicate]

Let $\mathbb{R}$ be equipped with the standard topology. Let $E$ be the set of irrational numbers. How do i prove that $E$ is not a countable union of closed subsets, using Baire Category Theorem?
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2answers
16 views

Set of equivalence classes equivalent to preimage of image of collapsing map proof

It says in a book I'm reading on topology that if $\mathit{R}$ is an equivalence relation on a space $X$, $p$ is the collapsing map $x \mapsto [x]$ and $A \subseteq X$ then: $$x \in A, y \mathit{R}x ...
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1answer
29 views

Number of topologies on 3 points

I have a computer program which tries to print all topologies on a finite set. This the output for $\{1,2,3\}$: ...
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3answers
43 views

Is it possible to have a continuous function $f : R → R$ and an open set $U ⊂ R$ with $f (U )$ is not open in $R$?

Is it possible to have a continuous function $f$: R → R and an open set U ⊂ R such that $f(U)$ is not open in R? If yes, could you provide an example.
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0answers
16 views

Help with some argument

Let $U$ be an neighborhood of $0\in{\mathbb{C}^2}$. And $K=\{(z_1,0):|z_1|<\rho\}$ be a subset of $U$, $L=\{z:|Rez_1^k|\leq \epsilon |z_1|^k\}$, where $\rho=\rho(\epsilon)$. Then i am supposed to ...
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2answers
21 views

A homomorphism induces a continuous map from ${\rm Spec}(A') \to {\rm Spec}(A)$.

Let $A, A'$ be commutative rings with $1 \neq 0$. Let $h : A \to A'$ be such that $h(1) = 1$. Then $f: {\rm Spec}(A') \to {\rm Spec}(A)$ defined by $f(\mathfrak{p}') = h^{-1}(\mathfrak{p}')$ is ...
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1answer
23 views

Homeomorphism between spaces equipped with cofinite topologies

I was given this question on my midterm. Currently I am studying for finals and am still unsure how to properly solve this question. Let X and Y be two sets and f be a map from X to Y be a bijection. ...
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2answers
46 views

Irreducible components of $Spec(A) $

A topological space $X$ is called irreducible if given $A_{1}, A_{2} $ open sets $ \neq \emptyset $ then $A_{1} \cap A_{2} \neq \emptyset$. The maximal irreducible topological subspaces of $X$ are ...
4
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1answer
32 views

Is the set of convex bodies include in a closed ball compact?

I consider the set $\mathcal{K}_B$ of convex bodies (convex and compact) which are include inside the unit closed ball of $\mathbb{R}^d$. I endow this set with the Hausdorff distance. Is it compact?
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2answers
58 views

Proving that a metric space with 3 points can be embedded isometrically into $\mathbb{R}^2$

My definition of an isometric embedding is that if $(M_2,d_1)$ and $(M_2,d_2)$ are metric spaces, then $G:M_1 \to M_2$ is an isometric embedding if $d_2(G(x),G(y)) = d_1(x,y)$ for all $x,y \in M_1$. ...
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1answer
21 views

define convergence for a sequence of elements in a metric space [on hold]

Define the convergence for a sequence of real numbers. How will you modify the definition to define convergence for a sequence of elements in a metric space? Give examples of convergent and non ...
2
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1answer
48 views

is a free group a discrete group?

Can I say that free groups are discrete groups? My question arises from the fact that free groups act on trees, and trees are graphs that can be viewed as a topological space (the graph topology).
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1answer
35 views

Distance to a set is continuous, revisited

In a metric space $(X,d)$, define $d(x,A) = \inf \{d(x,y):y\in A \}$. I am trying to prove that the set $A_{\epsilon} = \{ x \in X : d(x,A) < \epsilon \}$ is an open set for every $\epsilon>0$. ...
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1answer
35 views

How can I show that the interior of the set of interior points is an interior set? $(X^\circ)^\circ = X^\circ$

I am trying to show that the interior of a set of interior points is an interior point, that is, if $X$ is a subset of a metric space $M$, that $(X^\circ)^\circ = X^\circ$ for $X \subset M$. My ...
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0answers
30 views

The automorphism group of the real line with standard topology

How much is known about the automorphism group of the real line with the standard topology? Can it be described by a set of generators? I've been told that $\mathbb R$ has many weird and unexpected ...
3
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1answer
51 views

Specific Retraction from $\mathbb{R}^2$ to the logarithmic spiral

Let $X$ be a topological space. If $Y$ is a subspace of $X$, then $Y$ is a retract of $X$ if there exists a continuous function $r:X \rightarrow Y$ such that $r(y)=y$ for each $y\in Y$. The continuous ...
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0answers
42 views

On the Gromov-Hausdorff distance

I'm working on my bachelor thesis, and I'm studying principally on two textbooks (Selected Topics on Analysis in Metric Spaces [1] by Luigi Ambrosio and Paolo Tilli and A Course in Metric Geometry [2] ...
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4answers
83 views

Homeomorphism from $S^1$ to $\mathbb R$

I am trying to find a homeomorphism taking$$[0,1] \times S^1 \rightarrow \{ (x,y) \in \mathbb{R}: 1 \leq \|(x,y)\| \leq 2\}$$ I was thinking that a homeomorphism from $S^1$ to $\mathbb{R}$ gives the ...
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2answers
86 views

Homeomorphism Compact Subsets

Are there compact subsets $A,B \subset \mathbb{R^2}$ with $A$ not homeomorphic to $B$ but $A \times [0,1]$ homeomorphic to $B \times [0,1]$?
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1answer
57 views

The difference of closures is a subset of the closure of differences

I want to show that $\overline{A} \setminus \overline{B} \subset \overline{A \setminus B}$ holds for arbitrary sets in a topological space, and I am apparently always going the same wrong ways so is ...
7
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1answer
101 views

Isomorphism Finite Topological Space

Does there exist a finite topological space with fundamental group isomorphic to $\mathbb{Z_2}$?
2
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1answer
35 views

Proving that a point on the boundary of a closed ball in a metric space cannot be interior.

The idea of this proof is quite clear but I'm having some trouble making it rigorous. Suppose we have a metric space $(X, d)$ and a closed ball $U := \{x \in X : d(x, a) \leq t\}$ for some fixed $a$ ...
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1answer
94 views

How can a simple closed curve not look locally like the rotated graph of a continuous function?

A simple closed curve is a continuous closed curve without self-intersections. The question of whether you can inscribe a square in every simple closed curve is currently an open problem, but this ...
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2answers
27 views

the mapping class group of the disk is trivial proof

Proof : Identify $D^2$ with the closed unit disk in $\mathbb{R}^2$. Let $\phi : D^2 \rightarrow D^2$ be a homeomorphism with $\phi_{\partial D^2}$ equal to the identity. We define, $F(x,t) = ...
2
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0answers
31 views

Good topologies on $\mathcal{P}(X)$

Let $X$ be a topological space, and let $\mathcal{P}(X)$ (resp. $\mathcal{P}_0(X)$) be the set of all subsets of $X$ (resp. the set of all non empty subsets of $X$). Finally, let ...
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1answer
34 views

The only sets in $R^n$ which are both open and closed are the empty set and $R^n$ itself. [duplicate]

Proposition : The only sets in $R^n$ which are both open and closed are the empty set and $R^n$ itself. I came up with the proof of this claim and I'd like to know if my proof is correct. Proof: ...
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0answers
28 views

Computing fundamental groups.

How do I compute the fundamental groups of these spaces: (a) $\{(x,y)\in\mathbb{R}^2|x^2+y^2>1\}$; (b) $\mathbb{R}^2$ with two points deleted; (c) $\mathbb{C}$P$^n$, the complex projective ...
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2answers
32 views

Proof product topology has a countable basis

$(X,T_{X})$ and $(Y,T_{Y})$ are topological spaces We define the product topology as the family of unions of the sets in $\mathfrak{B}=\left \{ U \times V:U \in T_{X}, V \in T_{Y} \right \}$ If If ...
0
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2answers
39 views

Prove that the only sets in $R$ which are both open and closed are the empty set and $R$ itself. [duplicate]

I am trying to prove the above proposition. I tried to prove it by way of contradiction letting $S$ be such nonempty proper subset of $R$. Then $T=R-S$ would also be a nonempty proper subset of $R$ ...
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1answer
24 views

Totally disconnected orbit spaces

Let $X$ be a totally disconnected $G$-space, where $G$ is a locally compact Hausdorff group. Is the orbit space X/G also totally disconnected? The same question for locally compact, Hausdorff, ...
0
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1answer
18 views

Bases and open covers of a topological space

A basis of a topological space $X$ is a family of open sets ${B_i : i \in I}$ for some indexing set $I$, where any open set in $X$ can be written as the union of two or more members of ${B_i : i \in ...
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0answers
18 views

When a sort of weak topology is enough to generate vector space topology

Consider a vector space $V$, and some functions $f_\alpha: V \rightarrow \mathbb{C}$ where $\alpha$ ranges over some index set $A$. We can think about the coarsest topology which: a) makes the ...
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3answers
36 views

Compactness in $R^n$

I am looking at the proof that the following statements are equivalent from Apostol's Mathematical Analysis. Let $S$ be a subset of $R^n$. b) $S$ is closed and bounded. c) Every infinite subset of ...
2
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1answer
17 views

Show that $S^1 - \lbrace (1,0)\rbrace$ is homeomorphic to the open interval $(0,1)$

Be $S^1$ the unit circle in the plane, that is, $S^1= \lbrace (x,y) : x^2+y^2=1 \rbrace$ with the subspace topology. Show that $S^1 - \lbrace (1,0)\rbrace$ is homeomorphic to the open interval ...
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1answer
30 views

Subspaces and convergence in weak* topology

I would like to ask some questions regarding convergence in the weak* topology and subspaces. Let $X$ be a normed space with subspace $A \subset X$. Assume $X$ is endowed with the weak* topology. ...
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4answers
64 views

Topological spaces X and Y and a continuous bijection $f : X → Y$ while $f^{-1} : Y → X $ is not continuous

Give an example of topological spaces X and Y and a continuous bijection $f : X → Y$ such that $f^{-1} : Y → X $ is not continuous.
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1answer
26 views

The interval $(0,\infty)$ is an open set.

I want to prove this using interior points, $\epsilon$-neighborhoods and interior sets. The interior of a set A is denoted $A^o$. To show that $(0,\infty)$ is an open set, we must show that ...
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1answer
39 views

Ambiguity definitions - accumulationpoint

The literature is a bit ambiguous in my point of view. Limit points and accumulation points seems to be the same. I can accept that; that's just two names for the same. But I've seen different ...
3
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1answer
26 views

Alexander–Briggs notations for the links or knots of $N^3_m$

We can use Alexander–Briggs notations for the links or knots. For example, is three separate loops with no links. And there are many other examples of Alexander–Briggs notations for three ...
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1answer
21 views

Show that span is separable [on hold]

Let $X$ be a n.v.s and $A\subset X$. Show that if $A$ is enumerable then $\overline{ \text{span}\{A\}}$ is separable
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1answer
26 views

Is a mapping f bijection ?

If Y is an one point compactification of X,Y=X union {p}, p not belong to X. Is a mapping f from X into Y bijection? If it is not, what are the assumptions I add to be f bijection ? Thanks for any ...
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1answer
44 views

Is it true there exists $f:S^{2n}\longrightarrow S^{2n}$ making the diagram commutative?

Let $g:\mathbb R\mathbb P^{2n}\longrightarrow \mathbb R\mathbb P^{2n}$ be a continuous map where $\mathbb R\mathbb P^{2n}=\mathbb S^{2n}/\{\pm x\}$. Is it true there exists $f:\mathbb ...
2
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1answer
48 views

Covering map of $\mathbb R \mathbb P^2$

The question I am trying to answer is: Does the quotient map $ q:[0,1] \times [0,1] \to \mathbb R \mathbb P^2$ extend to a covering map $\mathbb R^2 \to \mathbb R \mathbb P^2$ I know that the ...
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0answers
26 views

Existence of particular open subgroups, given a prof-finite group

I have currently read a proof (existence of sections for pro-finite groups (in the book profinite groups of Ribes)) and I did not understand the following two facts used (without mentioning any ...