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; ...

learn more… | top users | synonyms (2)

2
votes
2answers
40 views

Show that the set is compact using the definition

The set in question is $\{0\}\cup \{1,\frac12,\frac13,\ldots,\frac1n,\ldots\}$ (for $n\in\mathbb N$). Okay, so for a set to be compact, every open cover of it must be able to be broken down into a ...
2
votes
1answer
47 views

Closed sets in a subspace are formed by intersecting the subspace with closed sets

Let $X$ be a metric space and let $Y$ be a subset of $X$ be a subspace with the induced metric. (induced means the metric restricted to elements of $Y$) Let $A$ be a subset of $Y$. Prove the following ...
1
vote
1answer
15 views

Isometric Operators: Common Core

Given a Hilbert or Banach space $\mathcal{H}$. Consider two closed operators $S:\mathcal{D}(S)\to\mathcal{H}$ and $T:\mathcal{D}(T)\to\mathcal{H}$. Suppose they're isometric on a common core ...
1
vote
2answers
91 views

When is the free loop space simply connected?

I am not sure if there is an obvious answer to this, but this has been bothering me. Let $X$ be a topological space. When is the free loop space, $LX$, simply connected? Correct me if I'm wrong, but ...
2
votes
1answer
17 views

Existence of maximizer implies compact? [duplicate]

I know that compact sets imply the existence of a maximizer, but is the converse true: Let $(X,d)$ be a metric space. Suppose that whenever $f$ is a continuous (and real) function on $X$, there ...
2
votes
1answer
42 views

MAth proof questions Open closed sets

Let $X$ be a metric space and let $Y$ be a subset of $X$ be a subspace with the induced metric. (induced means the metric restricted to elements of $Y$) Let $A$ be a subset of $Y$. Prove the following ...
1
vote
2answers
19 views

Open/closeness of subsets of natural numbers

So I've just started reading about neighbourhood and Hausdorff space. It makes me wonder if $(\mathbb{N},\mathcal{P}(\mathbb{N}))$ is Hausdorff and why, and are sets in $\mathbb{N}$ open or closed or? ...
1
vote
1answer
31 views

Limit points Topology

I'm trying to prove the following: Th: A subset of a topological space is closed iff it contains all of its limit points. Defn of a limit point of a subset $A$ is the following: $p \in X$ is a limit ...
1
vote
2answers
24 views

equivalent characterisation of simply connect spaces

I want to prove the following: Let $X$ be path connected space, $S^{1}$ the $1$-sphere and $D^{2}$ the unit circle. Following are equivalent: i)X is simply connected. ii) If $f:S^{1} \to X$ ...
1
vote
1answer
43 views

The Fundamental Group - An explicit homotopy between $(f \circ g) \circ h$ and $f \circ (g \circ h)$

I'm wondering if anyone can help me to understand a proof that the fundamental group is in fact a group. I am looking at the proof on page 3 of this document. I understand everything, although I am ...
2
votes
0answers
23 views

Lattice Version of Stone-Weierstrass

I've been reviewing Stone-Weierstrass theoerem. While reading the wikipedia page I read the following version of the theorem: Suppose $X$ is a compact Hausdorff space with at least two points and $L$ ...
1
vote
3answers
30 views

Topology-Open Sets of a Metric Space

Let $(X_i,d_i), i=1,2,\dots,n$ be metric spaces. Let $X=\prod_{i=1}^{n}X_i$ and let $(X,d)$ be the metric space defined in the standard manner. For $i=1,2,\dots,n$, let $O_i$ be an open subset of ...
3
votes
1answer
34 views

Tietze Extension Theorem

I saw Tietze extension theorem. Since its proof is non-trivial, I tried whether we can clarify it intuitively for functions of one real variable. So, in this special case, I am trying to prove that if ...
0
votes
0answers
12 views

what 2 by 2 matrix E subtracts the first component from the second component? [on hold]

what 2 by 2 matrix E subtracts the first component from the second component? E [3]=[3] [5] [2]
11
votes
2answers
104 views

Is the plane minus the integer lattice homeomorphic to the plane minus the integers?

The question, more rigorously posed, is: Is $\Bbb R^2-\Bbb Z^2$ homeomorphic to $\Bbb R^2-\Bbb Z\times\{0\}$? This question has been bugging me in the back of my head for a while now. Sometimes, ...
-1
votes
0answers
40 views

How could we define a sheaf or presheaf of polynomials? [on hold]

Good evening everyone , Is there a sheaf or presheaf whose sections are polynomials defined on opens of a topology ? . If yes , what is this topology ?. Is it the Zariski topology , and why? And how ...
0
votes
1answer
33 views

determine if set is open or closed

I have to determine whether the sset {1,2,3} is open or closed. I have never done these types of questions before but this is what I did (on pic). please can I have some feedback if I have done it ...
3
votes
1answer
65 views

Invariance of domain in $\mathbb{R}^2$

Let $U \subseteq \mathbb{R}^2$ an open subset and let $f:U\rightarrow \mathbb {R}^2$ is be a continuous function. I have the following version of Invariance of Domain Theorem (in $\mathbb{R}^2$): If ...
0
votes
0answers
11 views

Compact frames, an equivalent reformulation

$\top$ denotes the greatest element of a poset. Adapted from nLab: Definition 1. A frame is compact is and only if for every collection of opens whose union is $\top$ (which covers $\top$), there is ...
1
vote
1answer
39 views

A question regarding a discrete sub-space

denote by $A$ the set $\{\frac{1}{n} \,\, |\,\,\, n\in N \}$. Is $A$ a discrete sub-space of $\mathbb{R}$? (with the standard topology) I was told that it is, but I think it isn't. For example, ...
2
votes
0answers
18 views

finite simplicial complex compact

Let $K=(V,\Sigma)$ be a finite simplicial complex. I want to show that $|K|$ is compact. I know that $K$ is a sub-simplicial complex of $\Delta^V$ with $|\Delta^V|$ compact. So I think I should show ...
4
votes
1answer
29 views

What is the maximal size of an equal-distance set in $\mathbb{R}^n$?

Let $A\subseteq \mathbb{R}^n$ with the casual metric and $c\in\mathbb{R}^+$ be a real positive number, such that for every $a_1, a_2\in A$ if $a_1\neq a_2$ then $d(a_1,a_2)=c$. What is the maximal ...
3
votes
1answer
50 views

Is $\omega_1$ metrizable?

Following Urysohn's metrization theorem, I would like to prove or disprove that $\omega_1$ is metrizable. I know it is hausdorff, but I'm not sure whether or not it is second countable, and I'm at ...
3
votes
0answers
50 views

An open connected subset of compact metric connected and locally connected space, is path connected

I need to prove: If $X$ is compact, metric, connected and locally connected space, and $U$ is open connected subset of $X$, then $U$ is path connected. Using the following: a) If $X$ connected, ...
1
vote
1answer
28 views

The space of $S^1/S^1$, the space of a single point, and their first homotopy group

I read from the book Soft matter physics by Kleman that the space $R$ of a point is $0$ and its first homotopy group $\pi_1(0)=0$. This causes some confusion to my understanding. Why the space of a ...
2
votes
1answer
21 views

Why open unit ball in any infinite dimensional Banach space is finitely chainable?

In paper "Pointwise products of uniformly continuous functions" by Sam B. Nadler, Jr., He defined the finitely chainable as followings : Let $(X,d)$ be a metric space. An $\varepsilon$-chain in ...
4
votes
3answers
71 views

Involution on Cantor space with exactly one fixed point

Let $X=\{0,1\}^{\mathbb{N}}$ be the Cantor space. What is an example of a continuous map $\sigma : X \to X$ with $\sigma^2=\mathrm{id}$ and $\# \{x \in X : \sigma(x)=x\} = 1$? This has to exist, ...
3
votes
3answers
56 views

Proving that $ω_1$ is locally compact

I'm trying to show that $ω_1$ is locally compact, but when doing so, I need to show something else, which got me a bit stuck on. I'm taking a $\alpha\in ω_1$, so $\{\alpha\}$ is an open set. Since ...
6
votes
2answers
118 views

Covering $\mathbb{R}^2$ with uncountably many disjoint non-degenerate line segments

Is it possible to cover $\mathbb{R}^2$ with uncountably many disjoint non-degenerate line segments? If a formal definition is necessary, let's define a line segment as a set $\{(x, mx+c): x \in [a, ...
1
vote
1answer
18 views

Approximation of acontinuous function

How to approximate a continuous function on $[-\pi,+\pi]$ which is $2\pi$ periodic by a set of trigonometric polynomials in the sup-norm topology?
1
vote
1answer
28 views

Is there a quotient map between arbitrary topological spaces?

Let $X,Y$ be topological spaces. If there is a quotient map $p:X\rightarrow Y$, then the topology on $Y$ is completely determined by $p$. I'm curious whether the converse holds. That is, if we know ...
1
vote
1answer
45 views

Metric Space and Open Sets

I'm having trouble figuring out where to go with this problem. Any hints or strategies would be appreciated. I have just the basic definitions for open sets, distance metrics, etc. Consider $\Bbb ...
1
vote
1answer
25 views

Break up $\mathbb{R}P^2$ into a part homeomorphic to Mobius band & part homeo. to the 2-disc

The claim is that $\mathbb{R}P^2 = A \cup B$ where $A \simeq$ Mobius band, $B \simeq D^2$, and $A \cap B \simeq S^1$. I understand this intuitively with a gluing type argument, similar to the ...
1
vote
1answer
14 views

Continuity of function and its value.

Here's a problem I'm struggling with. Not really sure how to do this. My tools are epsilon delta proofs for continuity and that's about it. Let $f:[0,\infty)\to\Bbb R$ be a function which is ...
2
votes
0answers
32 views

Tangent bundles of smooth manifolds

Using the identity $T(M \times N) = T(M) \times T(N)$, it is easy to construct the tangent bundles for various smooth manifolds such as the n-dimensional sphere $S^{n}$. However, I could not figure ...
1
vote
2answers
49 views

If $P$ denotes the Cantor set, then show that $[0, 1] \setminus P$ is dense in $[0, 1]$

I have done the following. Suggest if the Proof is rigorous enough. Let us select any point $x\in [0,1]$. Ternary expansion of $x$ can be represented by $x=0.b_1 b_2 b_3 b_4\ldots$ where ...
2
votes
0answers
19 views

A question about the dimension of topological products

For each positive integer n, is the (small inductive) dimension of the topological product of n copies of the "long line", always equal to n? I ask because the "long line" is not a separable metric ...
1
vote
1answer
26 views

Convergence in $L^1 \cap L^2$

I am very confused about the following: Assume we have a sequence of functions $f_n \in$$L^1 \cap L^2 (\mathbb{R}^n)$. Then is it true that if this sequence is Cauchy both in $L^1$ and $L^2$, two ...
1
vote
0answers
53 views

Is the inverse limit of simplicial maps between finite directed graphs also a graph?

I think I have an intuitive understanding of why the following statement might be true, but I am not sure how to go about proving it. It's also possible my intuitive understanding is wrong and the ...
1
vote
1answer
18 views

Is there a bounded dense subset of norm linear space?

I have a question. In norm linear space $X$, we can find a bounded dense subset of $X$, can´t we?
3
votes
3answers
23 views

Prove that if $(X,d)$ is a compact metric space, and $K$ is an infinite set in $(X,d)$, then if $K$ has no limit point, $K$ is a closed set.

Prove that if $(X,d)$ is a compact metric space, and $K$ is an infinite set in $(X,d)$, then if $K$ has no limit point, $K$ is a closed set. Idea : Just like most topology proofs, the way I want to ...
2
votes
0answers
38 views

Operations on a smooth vector bundle

On a smooth vector bundle, one often defines addition and scalar multiplication to form a vector space. However, doesn't one need to show that these operations are smooth? Is this trivial or is there ...
1
vote
0answers
13 views

Topology of the intersection of toric arrangement

Hope someone will help me in the solution of the following question. I'm working on some topological problem involving the topology of the intersection of some characters of the torus. I want ot find ...
1
vote
0answers
6 views

Bounded uniform space

I studied that we do have a concept of total boundedness in a uniform space. I was thinking whether we have a concept of boundedness also in a uniform space (that need not be a metric space). Can ...
2
votes
1answer
36 views

Countable $T_2$-spaces that are invariant under point removal

Let's call a $T_2$-space $(X,\tau)$ invariant under point removal if for every $x\in X$ we have $X \cong (X\setminus\{x\})$ where $X\setminus\{x\}$ is endowed with the subspace topology. Examples of ...
1
vote
1answer
21 views

Initial topology

I have read that for all $U \in \sigma (X, {f_i, i \in I})$ (initial topology) there exists a finite number of open sets $V_i$ in $T_{Y_i}$ s.t. $U = \cap_{i\in G}f_i^{-1}(V_i)$ or $U = \cup_{i\in ...
4
votes
2answers
60 views

Prove that y>2x+1 is open?

The answer is inserted but what I'm looking for is a heavy breakdown on this. My professor tried to explain to me a harder version but I don't understand it. Solution: What my prof does for ...
3
votes
1answer
27 views

Show that the Sorgenfrey line does not have a countable basis.

I am trying to understand this proof from Munkres' book which shows that the Sorgenfrey line does not have a countable basis. His proof is: Let $\beta$ be a basis for $\mathbb{R}_l$. Choose for each ...
1
vote
1answer
43 views

This subcollection of the base is also a base

Let $B_n$ be a countable base of a locally compact Hausdorff space $X$. I am trying to show that $S = \{B_n \mid \overline{B_n} \text{ is compact} \}$ is also a base. I imagine the proof to be short ...
3
votes
2answers
163 views

Homeomorphic metric spaces

I want to examine if $(0,1] $ and $\mathbb R $ are homeomorphic. We work on metric space $(\mathbb R, e)$, where $e$ stands for the euclidean metric. My answer: Let's assume there is a ...