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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4answers
96 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
138 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
63 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 ...
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1answer
117 views

Isomorphism Finite Topological Space

Does there exist a finite topological space with fundamental group isomorphic to $\mathbb{Z_2}$?
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1answer
50 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
128 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
123 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) = ...
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0answers
43 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
49 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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2answers
36 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 ...
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2answers
78 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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2answers
60 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, ...
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1answer
39 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
25 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
41 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 ...
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1answer
37 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
51 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
92 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
43 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
43 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 ...
4
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1answer
64 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
36 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
52 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
57 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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1answer
55 views

Existence of particular open subgroups, given a profinite 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 ...
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1answer
65 views

Link between a topological space and a manifold

A topological space is defined as a non-empty set $X$ together with a given collection of subsets $T$ (topology) of $X$, such that, (i) any union of these subsets is one of the subsets. (ii) any ...
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1answer
55 views

weak-* topologies

Say $S = \{z \in \ell_\infty : z_n \in \{0,1\}\}$. Suppose I am asked a question about the weak-* topology on $S$. How am I supposed to make sense of this? The weak-* topology is a topology on a dual ...
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0answers
27 views

Can we say that $[0,\omega_1]$ is strictly-Frechet with no winning strategy in $G_{np}(q,E)$?

Let $E$ be a topological space, $q \in E$. The neighbourhood point game $G_{np}(q,E)$, is defined as follows. It is played by two players, ONE and TWO.In the n's step $n \in \omega$, ONE chooses ...
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0answers
26 views

Finding a uniformly continuous function from a non complete space to $\mathbb{R}^{+}$

Assume that $(X, d)$ is not complete. Prove that there exists a uniformly continuous function $f:X \rightarrow \mathbb{R}^{+}$ such that $\inf_{X} f(x) = 0$. We are guaranteed the existence of a ...
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2answers
30 views

Cantor Intersection Theorem in $R^n$

I am looking at the Cantor Intersection Theorem from Apostol's Mathematical analysis. Let {$Q_1, Q_2, $...} be a countable collection of nonempty sets in $R^n$ such that: 1) $Q_{k+1} \subset Q_k$ ...
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0answers
47 views

Showing $\mathbb{R}$ is a completion of $(\mathbb{Q}, | \cdot |)$

Recall the definition of a completion: A complete metric space $(Y, d)$ is said to be a completion of another metric space $(X, d)$ if there exists a map $f: X \rightarrow Y$ such that $f$ is an ...
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1answer
100 views

difference between sequence in topological space and metric space

Reading the book about topology, I find an interesting difference between two spaces: We use net convergence $\{p_\lambda\}_{\lambda\in\Lambda}\rightarrow p$ in frontier, but use sequence ...
2
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1answer
25 views

Inner product space is connected

How does one show any inner product space is connected? Shall I start with assuming that it is not connected, and arrive at a contradiction? So let $X$ be an inner product space, and there exist open, ...
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1answer
43 views

“internal” definition of complete regularity?

There is something strange (I think) about the complete regularity separation axiom. Consider the definitions. T0 means for every two distinct points there is an open set containing exactly one of ...
5
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2answers
98 views

A zero-dimensional Hausdorff space is totally disconnected

The full question: A space is zero-dimensional if the clopen subsets form a basis for the topology. Show that a zero-dimensional Hausdorff space is totally disconnected. Recall a space is totally ...
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1answer
44 views

Is there a subset of R such that their Cantor-Bendixson rank is the first limit ordinal?

I'm looking for a set $A \subset \mathbb{R}$ such that $\bigcap^\infty_{n=0} A^{(n)} $ is a perfect set (i.e $X'=X$) but $\forall n \in \mathbb{N}$ the set $A^{(n)}$ isn't perfect (where ...
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0answers
41 views

A lemma on function spaces

This is a lemma about function spaces. I'm not really understanding it however. Can someone try explaining it to me? Lemma: let $X$ be in |SET| $(Y, d)$ in |MET|, $f_n$, $f$ is in $Y^X$. Then $f_n\to ...
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0answers
28 views

Is the $C^r(M, N)$ space, with the strong (Whitney) topology, a Fréchet-Urysohn space?

Given smooth, non-compact manifolds $M$ and $N$, consider the function space $C^r(M, N)$. Equipped with the strong (Whitney) topology, this space is Hausdorff and Baire. It is, however, not first ...
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2answers
58 views

How do I prove this set is connected?

Define $A=\{(x,y):y=\sin(1/x), x\neq 0\}$ and $B=\{(0,y):-1\leq y \leq 1\}$. How do I prove that $A\cup B$ is connected? I can see this is not path connected but cannot prove why it is connected..
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1answer
86 views

Dense sequence in $[0,1]$

There is the theorem proved by Liouville which states that if $x$ is irrational then there are infinitely many fractions $\frac{p}{q}$ such that $|x-\frac{p}{q}|<\frac{1}{q^2}$, i.e. ...
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2answers
112 views

Limit of a sequence and a closed set

It's a dumb question, but I need to assure myself: If $V$ is a subset of a metric space $W$, then if we take a sequence in $V$ and it has a limit in $W\setminus V$, does it mean that $V$ is not ...
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1answer
46 views

Uniform convergence of sequence of functions on compact set [closed]

Let $X$ be a compact topologial space, $U$ an open subset of $\mathbb R$ and $f:X\to\mathbb R$ a continuous function such that $f(X)\subseteq U$. Prove that if a sequence of functions $f_n:X\to\mathbb ...
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0answers
65 views

Connected space whose every subspace is disconnected

We know that a subspace of a connected space can be disconnected eg. $\mathbf{Q} \in \mathbf{R}$ where $\mathbf{R}$ is connected but $\mathbf{Q}$ is totally disconnected as a subspace. My question ...
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0answers
65 views

$\mathbb{R}^n$ $\backslash$ $\mathbb{R}^k$ what does this mean?

$\mathbb{R}^n$ $\backslash$ $\mathbb{R}^k$ I saw this in my topology assignment. The question was about quotient spaces and homeomorphisms. I have never seen this expression before so it doesn't ...
5
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1answer
51 views

Topological Structure of Finite Set

I encounter with a problem in Topological Manifold written by Lee: How many different topological structure of $\{1,2,3\}$? It is easy to make a list of the question, and the answer is $9$. ...
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0answers
30 views

$L \cap V = \overline{L} \cap V \implies L$ open in $\overline{L}$. [Solved]

how do I prove this? My attempt: Let $x \in L$, and for some open set $V \ni x$ suppose that $L \cap V = \overline{L}\cap V$, also maybe use the fact that $L \cap V$ is closed in $V$. If $x \in L$ ...
0
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1answer
17 views

$L \cap V = \overline{L} \cap V$ when $L\cap V$ is closed in $V$.

Let $E$ be a topological space and $L, V \subset E$, $V$ open, and $L \cap V$ closed in $V$, then $\overline{L} \cap V = L \cap V$. Attempt: $L \cap V$ closed in $V$ implies $L \cap V = F \cap V$ for ...
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1answer
55 views

Path components of Wedge Sum

I couldn't find this anywhere else, so I decided to post it here. I suspect that the wedge sum $⋁X_α$ of pointed spaces $X_α$ has as path components all components of the topological sum $\oplus X_α$ ...
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0answers
56 views

Nagata Smirnov Metrization Theorem

I am looking for a proof for Nagata-Smirnov Metrization Theorem, but I couldn't find one that is readable. I found the paper by Nagata written in 1954 but it is unreadable and uses old notation. ...
0
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1answer
34 views

If a subset $E$ of $R^n$ is bounded then E is totally bounded

I am trying to prove the above proposition. The book that I am looking at contains E in a cube of the form T=[−b,b]×⋯×[−b,b] for some large b>0. Then, since any subspace of a totally bounded metric ...