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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224 views

Metrics and the Kuratowski closure axioms

Edit: Succinct proofs from user87690 can be found below, but I will gladly up-vote other valid approaches to any of the problems here! The following questions concern closure operators and the ...
4
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1answer
135 views

Axiomatic Bargaining: Nash's Solution

The following text is from the book: Bargaining and Markets by Osborne and Rubinstein, Academic Press Inc. Page 17 under the chapter The Axiomatic Approach: Nash's Solutions:. Two individuals can ...
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1answer
53 views

Is any closed set a derived set?

Is any closed set a derived set : in the real line ; in a finite dimensional vector space ; in an infinite dimensional vector space ? Thank you.
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3answers
47 views

Iteration of derived sets

Let $A$ be a set in the real line $\Bbb R$, and $A'$ the derived set of $A$, and $A''$ the derived set of $A'$, and so on. Is it possible to get an infinitively many distinct subsets of $\Bbb R$? ...
2
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1answer
230 views

Proper and free action of a discrete group

In Gallot, Hulin, Lafontaine's Riemannian Geometry: Definition Let $G$ be a discrete group, acting continuously on the left on a locally compact topological space $E$. One says that $G$ acts ...
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1answer
101 views

A property of compact subsets of metric spaces

Let $(X,\varrho)$ be a metric space and $K\subset X$ compact. Then, for every $\,\varepsilon > 0$, $\,K$ can be covered with a finite number of balls of radius $\varepsilon$. Show that the ...
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1answer
50 views

Question regarding notation in algebraic topology

My class has not been following a book and my professor's last bit of notation is a bit confusing to me. This is the goal. We are given a path-connected space $Y$ and $H$ a subgroup of $\pi_1(Y,y)$. ...
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3answers
56 views

Seeming ambiguity in the definition of open sets?

Concerning open sets: A set $S$ is "open" if and only if it is a neighborhood of each of its points. But for $S$ to be a neighbourhood of its points if there is some other set $V$ which contains an ...
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2answers
402 views

Why existence of universal covering implies that the base space be locally path connected?

I am reading Chapter 13, the chapter about classification of covering spaces, of J.Munkres' Topology. My confusion raised when I read Corollary 82.2. which says: the space $B$ has a universal ...
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0answers
60 views

$A$ is an interval so $A$ is connected?

I want to prove that if $A\subset \mathbb{R}$ is an interval then $A$ is connected. I found this proof, and I don't understand it essentially the ii) Suppose that $A$ is an interval but not ...
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0answers
28 views

How do you show Euler characteristic of any convex polyhedron is $2$?

In the Euler characteristic proof of a convex polyhedron, how do you show the cell decomposition of projection of two polyhedra 1) have a common refinement AND 2) that common refinement comes from ...
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1answer
86 views

Is this a topology?

Suppose that we have a set $S$ containing 0 and 1. Can we define our topology to be the four open sets $\varnothing$, $\{0\}$, $\{1\}$ and $\{0,1\}$? I know that the Sierpinski set contains the three ...
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1answer
46 views

Is $A$ compact, $f(A)$ uniformly continuous and is $f^{-1}$ continuous?

$X$ and $Y$ are metric spaces, $A\subseteq X$, $A$ is bounded. map $f:X\to Y$ is continuous. Questions: Is $A$ necessarily compact? Is $f(A)$ uniformly continuous? If given that $f$ is a ...
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0answers
211 views

Borel Measures: Atoms (Summary)

Disclaimer: The question here has been solved, now: Finest Measurable Partition (For jeapardy it is stated below, anyway. Have fun! ;) ) Summary: This is a summary of the discussions: ...
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2answers
385 views

On Equivalent Norms in an Infinite Dimensional Vector Space

How many non-equivalent norms can we define in an infinite dimensional vector space? Is there any explicit expression?
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1answer
60 views

Find a topological space X and a compact subset A in X such that closure of A is not compact.

Find a topological space X and a compact subset A in X such that closure of A is not compact. I first concluded that we must have X to be a non compact and a non Hausdorff space so that closure of A ...
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1answer
36 views

What does “the support of $f$ lies in $V$ mean?”

I have come across similar phrases and I am not sure what they mean. For example, if the phrase states "the support of $f(x)$ lies in a set $V$, does it mean that $V$ contains all $x$ such that ...
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0answers
24 views

Is local compactness preserved by continuous closed onto functions? [duplicate]

I've just shown for a homework problem that if $f$ is an open continuous function from $X$ onto a $T_2$-space $Y$, and $X$ is locally compact, then $Y$ is locally compact. I wonder, does this hold for ...
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2answers
417 views

If $X$ is locally compact, second countable and Hausdorff, then $X^*$ is metrizable and hence $X$ is metrizable

I need to show that: If $X$ is locally compact, second countable and Hausdorff, then $X^*$ is metrizable and hence $X$ is metrizable. I have already showed that every locally compact Hausdorff space ...
2
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3answers
72 views

Prove the set, {y ∈ X | r ≤ d(x,y) ≤ s}, is closed

Let r < s be positive real numbers and x ∈ X. Prove that the set: {y ∈ X | r ≤ d(x,y) ≤ s}, is closed. Having trouble with how I should tackle this ...
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2answers
21 views

Quick question: functions to spaces with equivalence relations

So I'm a little confused about sending functions from spaces without equivalence relations to a space with equivalence relations. For example, I'm trying to define a function $f : S^{n} \rightarrow ...
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1answer
41 views

Zariski-open subset in $\mathbb{C}^n$ to Zariski-closed subset in $\mathbb{C}^{n+1}$

Let $n \in \mathbb{N}$, $f\in \mathbb{C}[X_1,\dots,X_n]$, and $D(f):=\{x=(x_1,\dots,x_n)\in \mathbb{C^n}|f(x)\neq 0\}.$ I want to show that there is a injective $\Phi: D(f) \rightarrow ...
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1answer
52 views

What kind of Choice am I making in this argument?

I have an argument that's supposed to imply Choice, but I'm afraid it may be using some choice. If it does, how much choice? This is the part of the argument that might use some Choice. I marked the ...
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1answer
88 views

Cofinite Topology: Borel Algebra [closed]

Given the cofinite topology: $$\mathcal{T}:=\{U\subseteq\Omega:\#U^c<\infty\}$$ and generate its Borel algebra: $$\sigma(\mathcal{T})=\{E\subseteq\Omega:\#E\leq\aleph_0\lor\#E^c\leq\aleph_0\}$$ Why ...
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2answers
303 views

Borel Measures: Atoms vs. Point Masses

Let a measure be $\mu:\Sigma\to\mathbb{R}_+$. Call a measurable $A\subset\Sigma$ an atom if: $$\mu(A)>0:\quad\mu(E)<\mu(A)\implies\mu(E)=0\quad(E\subseteq A)$$ and a singleton $\{a\}\in\Sigma$ ...
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1answer
79 views

Retraction to an interval in a metric space

Suppose that $X$ is a metric space and $A$ is a subspace of $X$ that is homeomorphic to the interval $[0,1]$ with its usual topology. Let $v$ and end point of A. How do you proof that there is a ...
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2answers
384 views

Is continuity in topology well-defined?

In topology, a function is continuous if inverse of every open set is open. But for the inverse to be well-defined the function should be bijective. For example consider the projection map. It is not ...
3
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2answers
65 views

Applications of Baire's Threom [duplicate]

In a lecture on Baire's Theorem (for complete metric spaces), I gave, for a rather advanced undergraduate class in Real Analysis (covering the theory of metric spaces and elements of general ...
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4answers
94 views

Union of infinitely many closed sets

If $(K_i)_{i \in \mathbb{N}}$ is a sequence of closed sets in $\mathbb{R}^3$, then the union of these sets $\bigcup_{i=1}^\infty K_i = K_1 \cup K_2 \cup ... $ is also closed. My idea: ...
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1answer
83 views

Is this enough to prove a homeomorphism? — inverse on a dense subset

I want to prove that a map $f:A\to B$ is a homeomorphism, I know that $A$ is compact. I am not sure whether it is enough to show that: $f$ is continuous and injective for all $y\in B_1$, there is a ...
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1answer
65 views

Cantor's intersection Theorem without the diameter hypothesis

In proving Cantor's in intersection theorem, the fact that limit of the diameter of the sets is 0 was used to prove that the intersection is non-empty. I just wondered if that hypothesis is excluded ...
2
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2answers
108 views

Determining whether a set is open and bounded

I know that given $a < b$ and $g(x) \le h(x)$ $\{(x,y) \in \mathbb{R}^n |\ a \le x \le b, \ g(x) \le y \le h(x) \}$ is a closed constrained/bounded/limited (not sure what the terminology is in ...
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1answer
50 views

Euler characteristic of a convex polyhedron

In the Euler characteristic proof of a convex polyhedron, how can you show two cellular decompositions of two different polyhedron contain a common refinement?
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1answer
74 views

Cutting a torus enough times disconnects it

I am interested in showing that if you cut a torus too many times it becomes disconnected. Let $\mathbb T^n$ be the standard $n$-dimensional flat torus. Let $M_1, \ldots, M_k$ be $k$ disjoint smooth ...
4
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1answer
55 views

Universality with respect to quotients

Is there an infinite cardinal $\kappa$ for which the following statement (S) true? (S) : There is a topology $\tau_\kappa$ on $\kappa$ such that for all topological spaces $(X,\tau)$ with $|X|\leq ...
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0answers
63 views

Question about topological properties of $\Bbb{C}_p$

It is known that the structure of $p$-adic integers, $\Bbb{Z}_p$ is homeomorphic to the Cantor set, and $\Bbb{Q}_p$ is homeomorphic to the one-point deleted Cantor set (as I know, I don't certain it.) ...
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1answer
42 views

Sequence Lemma explanation

Then every neighbourhood $U$ of $x$ contains a point of $A$. So I don't see it happening unless $X$ is a metric space, but the proof is for any topological space.
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1answer
170 views

an interesting topology question about open sets

Suppose we are in $\mathbb{R}^n$ and say $\mathcal{B}$ is the collection of all open sets of $\mathbb{R}^n$ : all the open balls. we know $\mathcal{B}$ is a basis for $\mathbb{R}^n$. Now, put $$ T : ...
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0answers
73 views

Why do we care about non-$T_0$ spaces?

(Reminder: A $T_0$ topological space, also known as a Kolmogorov space, is a space where the topological structure "recognizes" that different points are different: No two points have exactly the same ...
4
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0answers
75 views

A proof of a small topological lemma

I just stumbled upon a proof of topological lemma that I don't understand: it would be great if anyone could give me some advices. To be blunt, I am convinced that the proof does work but to me it ...
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2answers
60 views

Exercise of topological spaces [duplicate]

$X$ is an infinite set and $T$ topology of $X$ in which all the infinite subset of $X$ are open, prove that $T$ is the discrete topology of $X$
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0answers
53 views

Tangent Bundles to manifolds

I am having trouble trying to visualize exactly what a tangent bundle to the klein bottle is spuposed to look like. Is it possible for one to decompose it as a direct sum of simpler bundles?
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1answer
36 views

Let $S_ 0$ be the space with $2$ points and the discrete topology. Find [$S_ 0$ , $X$] for an arbitrary space $X$.

Let $S_ 0$ be the space with $2$ points and the discrete topology. Find [$S_ 0$ , $X$] for an arbitrary space $X$. $[X,Y]=\{f:X\to Y,f$ continuous $\}/\sim$ where $\sim$ is the homotopic equivalence. ...
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1answer
15 views

Disconnecting a complex vector space

Can a (complex) dimension $n$ subspace disconnect a (complex) dimension $n+1$ vector space ? If the answer is no, what if we replace "vector space" by "manifold" ?
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1answer
46 views

Topology of metric completion of Euclidean metric

Lets consider $\cal{M}=\mathbb{R}^{2}\backslash\{(0,y)\}\text { with } \{|y|\le1\}$ with the Euclidean metric with line element $ds^{2}=dx^{2}+dy^{2}$. Now consider the distance function given by ...
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1answer
43 views

Prove that $C_1$ and $C_2$ are homotopic fixing endpoints.

Let $C_1$ and $C_2$ be two great circles in $S^2$, intersecting at the points $p,q$. If we consider $C_1$ and $C_2$ as curves starting and ending at $p$. Prove that $C_1$ and $C_2$ are homotopic ...
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1answer
32 views

The family of open intervals that do not contain $0$

Let $T$ be the collection of all open sets in $\mathbb{R}$ not containing $0$ union $\mathbb{R}$ i.e $$T=\{(a,b)\subset\mathbb{\bar R}:0\notin(a,b)\}\cup\{\mathbb{R}\}$$ Then what is true about $T$? ...
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0answers
58 views

Lattice representation of the Klein bottle

I'm looking at the space $\mathbb{R^2}/G$ where $G = \mathbb{Z^2}$ acts by $(n,m)(x,y) = ((-1)^mx+m,y+n))$ and I'm trying to show that this is a smooth surface. I am having a couple of problems. To ...
0
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1answer
35 views

Continuity of function proof

Let $f:X \to Y \times Z$ be given by $f(x)=\bigl( f_{1}(x), f_{2}(x) \bigr)$. Prove that $f$ is continuous iff $f_{1}$ and $f_{2}$ are continuous. I'm struggling to relate the pre image of $h$ ...
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1answer
78 views

String through 1 hole 3-torus

Okay So I had stayed up way too late thinking about this problem and I typed my question wrong. The question is: How do I deform a 3 dimensional 1 hole torus to go around a line? ...