Tagged Questions

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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0
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5 views

Cut sets in topology

As part of a problem sheet I have been asked to show that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$ whenever $n \neq m$. When I first proved that $\mathbb{R}$ is not homeomorphic to ...
1
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0answers
10 views

How to justify that $d_{\mid.\mid} ^{\alpha} (x,y) = \mid x-y\mid^{\alpha}$ (with $0<\alpha <1$) is a distance on $\mathbb{R}$

I know directly than : $\forall x,y \in \mathbb{R}$, $d_{\mid.\mid} ^{\alpha} (x,y)=0$ $\Leftrightarrow$ $x=y$ $\forall x,y \in \mathbb{R}$, $d_{\mid.\mid} ^{\alpha} (y,x) = d_{\mid.\mid} ^{\alpha} ...
0
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1answer
37 views

Partitions of $[0,1]$

Trying to test my understanding of analysis, today I came up with two questions, that will probably look obvious (be patience, because I am self-thaught). Anyway, here they are: 1) Is possible to ...
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2answers
29 views

Topology, locally compact

Let $E$ be a separable Banach space. Is $E$ locally compact space? I'm looking for a counterexample to this assertion. If you know anything please let me know.
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2answers
31 views

Closed sets with empty interior measure zero

Is the Lebesgue measure of a closed set with empty interior in $\mathbb{R}^{n}$ always zero? Trying to understand something in the math notes that I don't understand, and if the above is true, it ...
0
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0answers
20 views

Examples of compact subsets in topological spaces [duplicate]

I'm trying to find a topological space and a compact subset whose closure is not compact. It is an exercise in the text book Armstrong, Basic Topology, which I can not figure out. Any hint or answer ...
0
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4answers
30 views

How to find a bounded sequences with three sub-sequences that converge to three different limits.

The question is about finding a bounded sequence with three sub-sequences that converge to three different limits.
0
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1answer
60 views

$\Bbb R \times \Bbb R$ with a nonstandard topology

Let $\tau$ the topology on $\Bbb R \times \Bbb R$ generated by the collection of lines $y= 2x+k$ with $k\in \Bbb R$ (in the sense that the line $\{y=2x+k\}$ is a basic element for $\tau$) . Find ...
2
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1answer
64 views

Proof of $\Bbb R^n \not\eqsim \Bbb R$

Why is showing the same space but without a point is equivalent to show that the original spaces are not homemorphic? From what i could tell they just showed $\Bbb R - \{ x \} \not\eqsim \Bbb ...
1
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0answers
18 views

Understanding how connected sum of smooth surfaces is a surface

I have two smooth surfaces $M_1$ and $M_2$ I''m trying to understand how the connected sum $M_1 \mathop{\#} M_2$ is a smooth surface. I will write my understanding of the proof and then explain where ...
1
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1answer
19 views

How do I show using cut points that these two subsets of the plane are not homeomorphic?

How do I show using cut points or cut pairs that these 2 subsets are not homeomorphic? I cannot see any obvious cut points. I see that, in the first diagram the only cut pairs of type 1 when 1 ...
1
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3answers
43 views

Closure of this set is $\mathbb R^+$

Let $A$ be a subset of $\mathbb R^+$ which is not bounded. Prove that $\displaystyle \operatorname{cl}(\bigcup_{n\in\mathbb N^*}\frac{1}{n}A)=\mathbb R^+$ where $\frac{1}{n}A$ denotes ...
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1answer
30 views

Spectral Measures: Square Root Lemma

Given a Hilbert space $\mathcal{H}$. Consider a densely defined closed operator $A:\mathcal{D}(A)\to\mathcal{H}$. This gives rise to operators: $$A^*A:\mathcal{D}(A^*A)\to\mathcal{H}$$ ...
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1answer
26 views

How do I prove that a subset is closed in the topological space of $n \times n$-matrices.

Consider the topological space $M$ of $n \times n$ matrices over $\mathbb{R}$ equipped with the standard topology. Let $\mathcal{A} \subset M$ be the set of matrices such that $det(A) = 1$ for $ A ...
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0answers
44 views

Difficult question for me. Please help me to solve it [on hold]

I am trying to solve this question, but I couldn...
-4
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0answers
16 views

Homeomorphism between countable dense subspaces of R [duplicate]

I need to prove all countable and dense subsaces of $\mathbb{R}$ homeomorphic to each other.
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2answers
36 views

Is $\{(x,y)|y=\sin(\frac{1}{x})\}\cup(0,1)$ connected on $R^2$? [duplicate]

Source of definition: http://mathworld.wolfram.com/ConnectedSet.html Definition: A connected set is a set that cannot be partitioned into two nonempty subsets which are open in the relative topology ...
2
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3answers
493 views

Example of a countable open set of real numbers?

I am unable to think of a set that is both open and countable. While I can easily think of several that are not-closed and countable, finding explicitly open ones (besides the empty set) is proving to ...
0
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0answers
19 views

Homotopic attaching maps give Homotopy Equivalent spaces

I want to prove that if $f,g : S^{n-1} \to X$ are homotopic maps then the resulting spaces $X \cup_f D^n$ and $X \cup_g D^n$ are homotopy equivalent. I know this question has been asked before: ...
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0answers
14 views

Continuity definitions on non-compact subsets

At the top of the wikipedia articles on Hölder condition, Lipschitz Continuity and others, the chain of logic is as follows: On a compact subspace of a metric space: $$Continuously Differentiable ...
0
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0answers
21 views

$\ell_2^\mathbb{Q}$ as countable cartesian product

Since countable cartesian product of $0$-dimensional spaces (metrizable, separable) is $0$-dimensional, why $\ell_2^\mathbb{Q}$ being $1$-dimensional doesn't contradict that statement? ...
0
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1answer
17 views

Homeomorphism from $[0,1]\times[0,1]$ to $\overline{D}(0,1)$?

I'm trying to construct a homeomorphism from $[0,1]\times[0,1]$ to $\overline{D}(0,1)$. I'm pretty sure there is one. I've been trying to work geometrically : mapping $[0,1]\times[0,1]$ to ...
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2answers
44 views

Cantor Sets/nonempty/cardinality

Let $S_0=[0,1]$ and define every $S_k$ for $k\geq 1$ \begin{align*} S_1&=\left[0,\frac{1}{3}\right]\cup\left[\frac{2}{3}, 1\right],\\ S_2&=\left[0,\frac{1}{9}\right]\cup\left[\frac{2}{9}, ...
4
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1answer
26 views

Extension of the limit operator on $l^\infty$

Let $l^\infty = \{x\in \mathbb{R}^\mathbb{N}\colon \sup_{n\in \mathbb{N}}|x_n|<\infty\}$ and the subspace $C \subseteq l^\infty$ given by the convergent sequences. We consider the linear operator ...
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2answers
26 views

Convergence between weaker and coarser topologies of the same set

Suppose $X$ is a set with two topologies $S$ and $T$ and $S \subseteq T$. Prove or give counterexample : if ${x_n}$ converges in $(X,T)$ the it converge in $(X,S)$
6
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1answer
39 views

Can the order relation on $\mathbb{R}$ be recovered from the topology?

It's well-known that the usual topology on $\mathbb{R}$ is induced by the order relation, since the open intervals $\{(a, b) : a < b\}$ are a base for the topology. I am wondering if you can go ...
0
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2answers
118 views

why do you need to know topology to study differentional geometry

Why do I need to know topology to study differentional geometry? I just try to understand differentional geometry, but I am not sure why topology is needed for it. while I see that topology is an ...
2
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1answer
48 views

Metric spaces in topology

$(X,d)$ be a metric space. Given $\epsilon >0$ there is a non empty finite subset $X_\epsilon\subset X$ such that for every $x\in X$, we have inf$\{d(x,p):p\in X_\epsilon\}\leq\epsilon$. 1) Show ...
4
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0answers
46 views

Can open sets in different dimensions be homeomorphic to each other?

Assume $U$ is open in $\mathbb{R}^m$ and $V$ open in $\mathbb{R}^n$, $U\cong V$. Does it imply $m=n$?
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2answers
24 views

Formal proof that union of 2 non-disjoint open intervals is open interval.

Union of 2 non-disjoint open intervals is open interval. I will not use word open later. I want to obtain formal proof of this fact that is different from mine. My attemp: We know that union of ...
0
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0answers
21 views

Interior of sum of sets equals sum of interior of summands

I'd like to have the answer to the following question. If $X_1,X_2\subseteq \mathbb{R}^n$ are convex and compact sets of dimension $n$, does the following hold: ...
0
votes
1answer
32 views

Is true that $ \cap_{n=1}^{\infty} {\omega}_n \neq \emptyset $?

Let ${\omega}_n$ dense in $\mathbb{C}$ for all $n \in \mathbb{N}$, with ${\omega}_{n} \subset {\omega}_{n-1}$. Is true that $ \cap_{n=1}^{\infty} {\omega}_n \neq \emptyset $? If it true, is posible ...
0
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0answers
26 views

Help defining an open cover

Let $(x,y)$ be coordinates in $\mathbb{R}^2$. Let $A\subset \mathbb{R}^2$ such that $A=\left\{ y\geq 0 \right\}$. The following partition of $A$ is given: Let $0\leq K<+\infty$ \begin{equation} ...
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0answers
26 views

It is really difficult question for me. I tried to solve it, but I couldn`t. Please, help me to solve it [on hold]

Consider the topological space ([0,1),τ) , where τ=[0,y)|y∈[0,1] . For each of the following subsets, A of [0,1) , find A , A , and ∂A with respect to τ (Justify your answers in each case) (a)A ...
5
votes
1answer
73 views

Infinite removal of isolated points from a subset of $\mathbb{R}$.

Assume we have subset of $\mathbb{R}$. Then, at every step, we remove all isolated points from what is remained from initial subset. We stop when there is nothing to remove - so current set is either ...
1
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0answers
27 views

Proper Maps: Where is continuity used in this Wikipedia proof?

In this article on Wikipedia, a proof is given of the statement that any map $f$ from $X\to Y$ that is closed, continuous, and has the property that $f^{-1}(\{y\})$ is compact in $X$ for $y\in Y$, is ...
2
votes
3answers
71 views

Proof that every closed subset of $\mathbb R$ is finite or countable or continuum.

I want to prove that every closed subset of $\mathbb R$ is finite or countable or continuum. I know that for arbitrary subset we can not make similar statements - because of continuum hypothesis. ...
5
votes
2answers
104 views

Is $\mathbb{R}\setminus\mathbb{Q}$ a union of countable family of closed sets?

Can we represent set of irrational numbers as union of countable family of closed sets?
0
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0answers
35 views

On thinking that planarity is nothing but topology?

I've found the following quote on Harary's Graph Theory: And I'd like to know what it means. I know about the Kuratwoski theorem, which states that a graph is planar if no subgraph of it is ...
5
votes
0answers
43 views

“Wheel Theory”, Extended Reals, Limits, and “Nullity”: Can DNE limits be made to equal the element “$0/0$”?

"Wheels" are a little-known kind of algebraic structure: They modify the concept of a field or a ring in such a way that division by any element is possible, including division by zero, while also ...
1
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2answers
58 views

Homeomorphism and products

One of the notions that I have read about is that every homeomorphism is the finite product of homeomorphisms. I am trying to understand if it is possible to get a homeomorphism from non-homeomorphic ...
3
votes
1answer
59 views

shortest path in complete metric space

Let $(X,d)$ be a complete connected by arcs metric space. We define the length of a continuous path $\gamma: [0,1] \rightarrow X$ to be \begin{equation*} \sup\limits_{0=a_{0}<a_{1}<... a_{n}=1} ...
6
votes
1answer
41 views

A space which is not compact but in which every descending chain of non-empty closed sets has non-empty intersection

A topological space $X$ is compact if and only if any collection of closed sets satisfying the finite intersection property has non-empty intersection. Clearly, this implies that compact spaces $X$ ...
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votes
0answers
52 views

2 difficult questions [on hold]

$1.$ Consider the topological space $([0,1), \tau)$, where $\tau = {[0,y)|y \in [0,1]}$. For each of the following subsets, $A$ of $[0,1)$, find $A$, $A$, and $ \partial A$ with respect to $\tau$ ...
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0answers
23 views

Homeomorphism between open subset and cartesian product

Let $U$ be an open set of metrizable space $(X,\cal{T}_X)$. I am asked to prove that $U$ is homeomorphic to every closed subspace of cartesian product $X\times\mathbb{R}$, where $\mathbb{R}$ is given ...
0
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0answers
33 views

Show that domains in $\mathbb{R}$ are exactly open intervals $(a,b)$

Show that domains in $\mathbb{R}$ are exactly open intervals $(a,b)$ I would like to show that domains in $\mathbb{R}$ are exactly open intervals $(a,b)$ where $a,b \in \Bbb{R}$. Can I say that I ...
0
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1answer
29 views

Proof of theorem 28.13 in “General Topology” by Willard

From Willard S. General Topology. I think I'm missing something. In the function $F$, how does the value of $F(x)$ determined for $x\in K\setminus D$ such that $x\neq a,b,p$? I get that $F$ extend ...
1
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2answers
68 views

Showing $F^{-1}(C)$ is compact when $C$ is compact.

$f : X → Y$ is a map. If f is closed, and $f^{−1}(y)$ is compact in $X$ for each $y ∈ Y$ then show that $f^{−1} (C)$ is compact in $X$ for any compact subset $C$ of $Y$ . How does the proof go ...
0
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1answer
22 views

How do I show that an element of a topological space lies in the closure of a subset of a topological space?

Consider the vector space of functions $X = \{ f : \mathbb{R} \to \mathbb{R} \}$ with the topology $\tau$ generated by a subbasis given by $S_{n,u,v} = \{f \in X | u < f(n) < v\}$. Let ...
0
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0answers
44 views

Question about a solution to a problem in Hatcher

I'm reading through a solution to a problem in Hatcher and I'm not sure why this line is true. Could anyone explain?