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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23 views

Understanding continuity in metric space

Let $(X,d)$ be a metric space, $x_0 \in X$ and $r>0$ a real number. The open ball in X of radius r centred on $x_0$ is the set $B_r (x_0)=\{x \in X : d(x,x_0) <r \}$ The notation if ...
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1answer
56 views

The Klein bottle is homeomorphic to the boundary of the product of the Möbius band with a disk

Can someone please give me a hint or the intuition in how to prove that the Klein bottle $\cong \partial$(Möbius strip $\times D^1 $ ) where $\cong$ means homeomorphic.
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1answer
19 views

Understanding open balls in metric spaces

Let $(X,d)$ be a metric space, $x_0 \in X$ and $r>0$ a real number. The open ball in X of radius r centred on $x_0$ is the set $B_r (x_0)=\{x \in X : d(x,x_0) <r \}$ Let $X=\mathbb{R}^2$ and ...
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51 views

Good transition books

I am finishing up a class on discrete mathematics and I am interested in skipping my schools transition courses in order to take a rigorous theory course next semester (topology, analysis, abstract ...
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49 views

Prove the product topology is the equivalence of metric topology in a special case.

Define a topology T on D$^\Bbb N$ with D={0,1} and the discrete metric as the following: C $\in$ T is basic iff there's an i $\in \Bbb N$ such that $C := C_1 \times C_2 \times...\times C_j \times D ...
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1answer
36 views

In a Topological vector space, a subspace of codimension 1 is either dense or closed.

Let $X$ be a topological vector space and $V$ be a linear subspace of $X$ such that $\text{dim}(X/V)=1$, then either V is closed or $\overline{V}=X$. In other words if it is not closed then it ...
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1answer
15 views

Collection of all open sets whose closure is contained in a cover is a cover (of a regular space)

Let $X$ be a regular space. Let $\mathcal{A}$ be an open cover of $X$. We define $\mathcal{B}$ as the collection of all open sets $U$ such that $\overline{U}$ is contained in an element of ...
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56 views

How well path-connected can a (special) partition of $\Bbb R^2$ be?

It turned out that How well connected can a (special) partition of $\Bbb R^2$ be? had a a few nice answers using continum-many pairwise disjoint dense connected sets. Connectedness is just weirder ...
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36 views

Given $(X,A)$,$(Y,B)$ such that $X/A$ and $Y/B$ are homotopy equivalent,are their relative homology groups isomorphic?

Suppose that $(X,A)$,$(Y,B)$ are pairs of topological spaces. If $X/A$ and $Y/B$ are homotopy equivalent, are $H_*(X,A;\mathbb{Z})$ isomorphic to $H_*(Y,B;\mathbb{Z})$ ?
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43 views

An exercise related to Krull topology - showing that two bases define the same topology

The following is a definition from my lecture notes: Let $L/K$ be a Galois extension with $G=Gal(L/K)$ then the family subgroups $Gal(L/L_{i})$, where $L_{i}$ runs over all finite ...
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1answer
38 views

compactness of the Stone-Čech compactification by ultrafilters

My question is about the proof of compactness of the Lemma 3.1(page 5) in this paper. Let $\beta \mathbb{N}$ be the set of all the ultrafilters on $\mathbb{N.}$ For each $A\subseteq \mathbb{N}$, we ...
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1answer
16 views

A homeomorphism of B^n fixing the boundary?

I am trying to construct an automorphism $$\phi:\mathbb{\overline B}^n\to\mathbb{\overline B}^n$$ such that $\phi(0) = \alpha\hat x_1$, and $\phi|_{\partial\mathbb{\overline B}^n} =$ Id. I thought ...
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1answer
40 views

Show the relationship between a compact(non empty) set $A$ and a sequence $A_j$ in $R^n$:

Let $A_j$ $(j\in N)$ be a sequence of non-empty compact subsets of $R^n$, and $A$ is a non-empty compact set, when $A_j$ converges in the Hausdorff metric to $A$, I need to show that: $$ A=\overline ...
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1answer
51 views

Topological Proof of Infinite Primes

Suppose $\mathbb{Z}$ had the topology where every arithmetic sequence is open. Each arithmetic sequence is also closed. I showed that every non-empty open set is also infinite. My question is in the ...
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1answer
41 views

Diameter of set ${(a,b):0\leq a,b\leq 2}$ on $\mathbb{R}^{2}$

Set $A=\{(a,b):0\leq a,b\leq 2\}$, $a$ and $b$ both belong to $\mathbb{R}$. What's the graph of set $A$ and what's the diameter of it? I have the answer given on my answer sheet that the diameter is ...
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2answers
21 views

Subsets of a Cartesian product are disjoint iff there exist projections that are disjoint

The Theorem? Suppose $\{X_\alpha\}_{\alpha \in J}$ is a family of non-empty sets. Let $X = \prod_{\alpha \in J} X_\alpha$. For each $\alpha \in J$ define $p_\alpha : X \to X_\alpha$ to be the ...
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1answer
55 views

$C_{c}(X)$ is complete. then implies that $X$ is compact. [closed]

Let $X$ is locally compact Hausdorff space .If $C_{c}(X)$ is complet,then $X$ is compact (this is to be proved). I know that $C_{c}(X)$ is dense in $C_{0}(X)$. As $C_{c}(X)$ is complete implies that ...
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0answers
20 views

$C_{c}(X)$ is complete implies $X$ is compact. [duplicate]

If $X$ locally compact Hausdorff space. Then $C_{c}(X)$ is complete implies $X$ is compact. I know that $C_{c}(X)$ dense in $C_{0}(X)$. So in that case $C_{c}(X)=C_{0}(X)$. I know only Tiez ...
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1answer
78 views

Is there a surjective continuous $f : [0,1] \rightarrow {[0,1]}^{2}$.

I know that There is such a function like this but the domain is cantor set.
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2answers
50 views

Counterexample to show homeomorphic spaces don't need to have the same dense subsets

Given a set $X$ with two homeomorphic topologies $\mathcal{T}_1$ and $\mathcal{T}_2$ on X. Consider the statement: A subset $U \subseteq X$ is dense in $\mathcal{T}_1$ iff $U$ is dense in ...
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1answer
43 views

Union of closed balls centered with centers in a closed set is closed in Euclidean space

Let $\mathbb{R}^n$ be the Euclidean space with the euclidean metric $d(x,y)=\|x-y\|_2$. Let $A$ be a closed subset of $\mathbb{R}^n$ and for each $a\in A$, $D_a$ is a closed ball centered at $a$ such ...
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1answer
94 views

Is the graph of $f(x)=\sin(\frac 1x)$ closed?

Let $f:X\to Y$ be a function in the metric spaces $X,Y$. The graph of $f$ is the set: $$G(f)=\{(x,f(x));x\in X\}$$ And we say that the graph of $f$ is closed, if the set $G(f)$ is closed in $X\times ...
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1answer
33 views

Closed ball is a closed set by showing that it's equal to its closure

Given the Euclidean space ($\mathbb{R}^n, d_{\|\cdot\|_2}$), $x \in \mathbb{R}^n$ and $r >0$, I want to show that the closed ball $B = \overline{B_{d_{\|\cdot\|_2}}(x,r)}$ is closed by showing ...
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0answers
28 views

Proof that countable compliment topology is a topology

I have the definition of countable compliment topology as : X is any non-empty set and τ = τc is the collection of all subsets U such that $U^{c}$ = X\U is either countable or all of X. How do I ...
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1answer
48 views

Classification of Polish topologies on a countably infinite set

Let $X$ be a countably infinite set. While investigasting the literature on Polish spaces, I met so far only examples for compact or locally compact Polish topologies on $X$: the order topology on ...
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3answers
48 views

Show that set is open

$E=\{(x,y):x^2+2y^2<6\}$ Let $X=(x,y)$ how can I find radius $ $ $r>0$ such that $B_r(X)\subset E$ for all $X\in E$ ?
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1answer
57 views

If $X=\{1,2,3\}$, is $\tau=\{\phi,\{0\},\{1,2\},X\}$ a topology on X?

I have been told that if $$X=\{1,2,3\}$$ and $$\tau=\{\phi,\{0\},\{1,2\},X\}$$ then $\tau$ is a topology on X. But I am not sure why. The first axiom is held, $\phi$ and $X$ are in $\tau$. Now I ...
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0answers
24 views

Set operations in different dimensions

I was wondering, whether there exisits a globally fixed definition of elementary set operations of convex sets of different dimension. To be more specific: Thinking of the intersection of 2 boxes (a ...
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1answer
39 views

Set in $\mathbb{R}$ with no open intervals.

Let $X$ be a compact set of $\mathbb{R}$, containing no open intervals. Does it follow that the set is totally disconnected?
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1answer
34 views

SO(n) homeomorphic to cartesian product of spheres

I suspect that $O(n)$ is homeomorphic to a product of spheres $S^m$ (equipped with the product metric) for various $m$ like so: $$O(n) \cong S^{n-1} \times S^{n-2} \times \dots \times S^0$$ I need ...
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The quotient of two metrics may not be a metric

I'm looking for a counterexample to show that $\frac{\rho_1}{\rho_2}$ is not necessarily a metric where $\rho_1$ and $\rho_2$ are metrics. We define $\frac{\rho_1}{\rho_2}$ as follows: ...
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1answer
23 views

Comparing Topologies (Product and cofinite)

I have a midterm coming up and would love some help/guidance (or solution) on this practice problem. Let X be an infinite set with the finite complement topology. Let T1 denote the product topology ...
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1answer
34 views

A subset of $\mathbb{R}$ is compact iff it is sequentially compact

Let $A \subset \mathbb{R}$. Show that $A$ is sequentially compact if and only if $A$ is compact. I have looked for other explanations of this, but I can't find one that is for $\mathbb{R}$ ...
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1answer
25 views

If Sequentially closed then closed

If $A \subseteq \mathbb{R}$ is sequentially closed then it is closed. I'm trying to prove this directly. Since $A$ is sequentially closed then if $(a_n)$ is a sequence in $A$ converging to a limit ...
2
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1answer
38 views

Can every map between manifolds be factored as $p\circ i$

Can every map between topological manifolds $f\colon X\to Y$ be factored as $p\circ i\colon X\to \overline{Y}\to Y$ with $i$ inclusion of open subset in another topological manifold $\overline{Y}$ and ...
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1answer
35 views

Show that $X = \{ \text{The set of all series which converge to } a \in [-1,1] \}$ is not compact.

Show that $X = \{ \text{The set of all series which converge to }a \in [-1,1] \}$ is not compact. The metric on this space is given by $$ d\left(\bar a, \bar b\right)=\left(\sum_{k=1}^{\infty}| a_k ...
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1answer
90 views

Munkres exercise

Problem Let $\{A_\alpha\}$ be a collection of subsets of $X$; let $X=\bigcup_{\alpha}A_\alpha$. Let $f:X\rightarrow Y$;suppose that $f\vert_{A_\alpha}$ is continuous for each $\alpha$. An index ...
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1answer
152 views

Why are eigenfunctions of Laplace-Beltrami operators the minimizer of $\int_\mathcal{M}\| \nabla f(x)\|^2$?

Given a smooth $m$-dimensional manifold $\mathcal{M}$ embedded in $\Re^k$. Suppose we have a map $f : M \to \Re .$ Now, these are my questions: Specific question: i): Why does the $f$ that ...
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1answer
35 views

Open ball around a set

I'm reading "Topics on Continua" by Sergio Macías, and he provides the following notation on page 2: "If $\varepsilon$ is a positive real number, then the symbol $\mathcal{V}_{\varepsilon}^d(A)$ ...
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1answer
35 views

Banach Fixed Point Theorem and the function $f(x)=x+e^{-x}$

let $X = [0,\infty)$ equipped with the standard metric $d(x,y) = |x-y|$. Let $f: X \rightarrow X$ be defined by $f(x)=x+e^{-x}$ Explain why this function doesn't contradict Banach's Fixed Point ...
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0answers
18 views

Metric space and continuity proof. [duplicate]

I have this proposition. Let X,Y be a metric spaces and let $f:X \to Y$ be a map. Then f is continuous if and only if $f^{-1} (V)$ is closed in X whenever V is closed in Y. How would you go about ...
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2answers
43 views

Find two open intervals A,B⊂R such that (A∩B) closed ≠A closed∩B closed

 Hi, I cannot come up with two open intervals in the reals that show the closed intersection of two sets does not equal the intersection of two closed sets. Thanks! 
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1answer
29 views

Action induces action of group ring on singular chain complex. [duplicate]

See here for a related question. Let $X$ be a space that satisfies the hypotheses used to construct a universal cover $\overline{X}$. Let $\pi = \pi_1(X)$ and consider the action of the group $\pi$ ...
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1answer
40 views

simply connected/path connected of a set in $\Bbb R^2$

Is the set $\{(x,y)\ |\ x^2-y^2<0\}$ in $\Bbb R^2$path connected? Is it simply connected? I am having trouble with these concepts in these scenario. Any help would be appreciated.
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1answer
59 views

Confusion on the meaning of the Vitali- Covering Lemma

In Folland, Lemma 3.15 gives a version of the Vitali-Covering Lemma: Lemma 3.15: Let $\mathscr{C}$ be a collection of open balls in $\mathbb{R}^{n}$, and let $U= \bigcup_{B \in \mathscr{C}} B$. If ...
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3answers
103 views

How to prove that $\left(\overline{A}\right)'\subset A'$ in a Hausdorff space

If $E$ is a Hausdorff space, and $A\subset E$ how to prove that $\left(\overline{A}\right)'\subset A'$ ? We say that $x\in A'$ if and only if $\forall V\in \mathcal{V}_x, (V\setminus\{x\})\cap ...
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0answers
50 views

Segment ordered density conjecture revisited

I have a set $S\subset\mathbb {R}^2$ with the following property (P) $\forall x,y\in S$, $\forall\mathscr{C}$ a convex set that contains $x$ in its interior, $bd\mathscr{C}\cap [x,y]\subset ...
2
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2answers
60 views

How well connected can a (special) partition of $\Bbb R^2$ be?

Let $\{A_i\}_{i\in I}$ be a family of subsets of $\Bbb R^2$ (where $I=\Bbb N$ or $\Bbb Z$; I don't know if it makes a difference) such that $\bigcup_{i\in I} A_i=\Bbb R^2$ $i\ne j\implies A_i\cap ...
2
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1answer
122 views

Homemorphism from $S^n$ to $S^n$

Let $S^n$ be the unit $n$-sphere($n\geq2$) and $X=\{a_1,...,a_k\}$, $Y=\{b_1,...,b_k\}$ be two finite subsets of $S^n$, does there exist a homemorphism $f$ from $S^n$ to $S^n$ such that $f(a_i)=b_i$ ...
2
votes
2answers
102 views

Proof of non-existence of a continuous bijection between $\mathbb{R}$ and $\mathbb{R}^2$

There are a lot of websites and forums, which explain that there is a bijection between $\mathbb{R}$ and $\mathbb{R}^2$, and even give some bijections. (By the way: Can you generalize it? since it ...