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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3
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2answers
75 views

Is a set closed if it has no accumulation points?

I was wondering if a set $A$ has no accumulation point, is this set $A$ closed? I think this is true, but I'm not quite sure. Here's my thinking: By closed set definition: A set $A$ is closed ...
4
votes
0answers
26 views

Show that $S^n\cong\mathbb{R}^n\cup\{\infty\}.$

The problem statement is: Show that $S^n\cong\mathbb{R}^n\cup\{\infty\}.$ My attempt at the proof is as follows: Let $f:S^n\to\mathbb{R}^n\cup\{\infty\}$ be defined as $f(x)=h(x)$ for $x\neq p$ and ...
0
votes
0answers
16 views

Topology on $C_{compact}^{\infty}(R)$

Want to show that the topology on $C_{\mathrm{compact}}^{\infty}(R)$, which is given by all the good semi-norms, is generated by the following collection of semi-norms $\| \cdot\|_{m,\epsilon}$ ...
4
votes
2answers
40 views

How to prove $D^2\setminus\{0\}$ is not homeomorphic to $\mathbb{R}^2\setminus\{0\}$?

Here $D^2$ denotes the closed unit disk in $\mathbb{R}^2$. I know that $D^2$ is not homeomorphic to $\mathbb{R}^2$ as $D^2$ is compact. Intuitively I believe that $D^2\setminus\{0\}$ is not ...
0
votes
1answer
31 views

Is this feature of the product topology still true if we take product to infinity?

We have been asked to show "Let $X_1, \ldots, X_n$ be topological spaces. Show that the product topology is the unique topology on $X_1 \times \cdots \times X_n$ with the property that, for any ...
6
votes
2answers
77 views

Is every open set a continuous image of a closed set? (in Euclidean space)

Let $A \subset \mathbb{R}^n$ be an open subset. The question is whether $A$ can always be written as a continuous image of a closed subset of euclidean space $f(C) = A$ for some closed ...
0
votes
1answer
23 views

Two proofs of cartesian product in topological space

Let $(X \times Y, \tau)$ be cartesian product of topological spaces $(X, \tau_X)$, $(Y, \tau_Y)$. Let $ A \subset X$, $ B \subset Y$. A) Prove that $\overline{A\times B}= \overline{A} \times ...
1
vote
0answers
23 views

Euler characteristic of non-convex polyhedra

Why is the euler characteristic of non-convex polyhedra $\neq 2$ always? 1) If one removes an edge do we lose one face as in the convex case? And so too removing a vertex that bridges two edges ...
0
votes
2answers
22 views

A problem of topology about connectedness

Consider a topology $T$ on $\mathbb{R}$:{The all open sets in $\mathbb{R}$ not containing $0$} union { $\mathbb{R}$}.Then which of the following is true? $1.T$ is connected. $2.T$ is hausdorff. I ...
1
vote
1answer
26 views

Question about limit point

I have this set $A=\lbrace (-1)^n(1+\frac1n), n\in \mathbb{N}\rbrace$ and i m looking for the limit point of $A$, i take $u_n= (-1)^n(1+\frac1n)$ and i found that $\lim_{n\rightarrow\infty}u_{2n}=1$ ...
0
votes
2answers
38 views

how to show a set is path connected?

Say $X=\{a,b,c,d\}$ and $\tau=\{\emptyset,X,\{a\},\{c\}, \{a,b,d\}, \{b,c,d\}, \{b,d\}\}$ This is a topological space as it satisfies the appropriate properties but how do you show $\tau$ is ...
1
vote
1answer
18 views

Homology groups of three faces with a point on the common edge removed

Consider this situation: There is an edge between two vertices, with three faces (maybe half-disks or half-squares, it doesn't really make a difference to topology as far as I know) going out from it, ...
0
votes
1answer
17 views

Relation between connected subset and clopen subset of a metric space?

I've read that for $A$ a connected subset of a metric space $M$ and $C$ a clopen (closed and open) subset of $M$, one could prove that either $A \subset C$ or $A \cap C=\varnothing$ and use it to ...
1
vote
0answers
16 views

condition for homeomorphism

If $X $ and $Y $ are homeomorphic as topological spaces is there any necessary and sufficient condition for $X\setminus A$ and $Y \setminus B$ to be homeomorphic?$ A\subseteq X ,B\subseteq Y$
0
votes
2answers
20 views

Metrics and the Kuratowski closure axioms

The questions below concern closure operators in the sense of the Kuratowski closure axioms. Additional tags and clarifications for the problems are all welcome. Fact: Let $X$ be a nonempty set ...
1
vote
0answers
18 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 ...
0
votes
1answer
21 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.
1
vote
3answers
23 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 $\Bbb R$? ...
-1
votes
1answer
34 views

A closed set is a derived set, true or false? [on hold]

My question is on the title : true or false : a closed set is a derived set ? Thank you very much.
2
votes
1answer
29 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 ...
0
votes
1answer
50 views

A property of compact sets

Let $K$ be a compact set. Prove that for all $\,\varepsilon > 0$, $\,K$ can be covered with a finite number of neighbourhoods of radius $\varepsilon$. Show that the reciprocal is not true. My ...
2
votes
1answer
42 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 $\pi_1(Y,y)$. ...
1
vote
3answers
36 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 ...
1
vote
2answers
34 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 ...
1
vote
0answers
38 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 ...
1
vote
0answers
22 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 ...
0
votes
1answer
59 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 ...
0
votes
1answer
25 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 ...
0
votes
0answers
46 views

Is $\mathbb{R}^n$ not nowhere dense? [on hold]

Is $\mathbb{R}^n$ not nowhere dense? I am trying to show that a clopen set is not necessarily nowhere dense. I know that it is both open and closed, but I am not sure how to find the interior of ...
0
votes
0answers
27 views

Borel Measures: Atoms (Summary)

Disclaimer: This is a summary of the discussions: Measure Atoms: Definition? Borel Measures: Discrete & Continuous? Borel Measures: Atoms vs. Point Masses Reference: Further results are ...
2
votes
2answers
45 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?
1
vote
1answer
43 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 ...
0
votes
1answer
31 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 ...
0
votes
0answers
10 views

$\overline{M\setminus X}=M\setminus\operatorname{int}(X) $? [duplicate]

Why is true that for $X\subset M$ (where $M$ is a metric space), we have that $\overline{M\setminus X}=M\setminus\operatorname{int}(X) $? Not sure why.
1
vote
0answers
19 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 ...
3
votes
2answers
30 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
votes
3answers
29 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 ...
1
vote
2answers
17 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 ...
0
votes
1answer
34 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 ...
1
vote
1answer
44 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 ...
0
votes
0answers
21 views

Power sets and Discrete Topologies.

These are the definitions I have learnt; Let $X$ be any non-empty set and $\tau$ be the collection of all subsets of $X$. Then $\tau$ is called the discrete topology on the set $X$. The ...
0
votes
0answers
15 views

Cofinite Topology: Borel Algebra?

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 ...
4
votes
2answers
78 views

Borel Measures: Atoms vs. Point Masses

Let a measure be $\mu:\Sigma\to\mathbb{R}_+$. Call a measurable $A\in\Sigma$ an atom if: $$\mu(A)>0:\quad\mu(E)<\mu(A)\implies\mu(E)=0\quad(E\subseteq A)$$ and a singleton $\{a\}\in\Sigma$ a ...
1
vote
1answer
46 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 ...
3
votes
2answers
285 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
votes
2answers
27 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 ...
2
votes
4answers
30 views

Union of infinite 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: ...
1
vote
1answer
26 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 ...
0
votes
1answer
29 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 ...
1
vote
2answers
30 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 ...