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)

0
votes
1answer
19 views

Equivalence between two topological statements concerning the basis of a topology.

I need to show the following statement Let $\mathcal{B}\subset P(X)$ be a set of subsets of a set $X$, such that $\bigcup_{U\in \mathcal{B}}U =X$ then the following are equivalent $i)$ there ...
0
votes
0answers
18 views

Proof related to Tychonoff's Theorem

A proof related to Tychonoff's Theorem is that the intersection of a finite number of members of the maximal class M also belongs to the class M.  (See for example problem 7 p175 in Lipschutz's ...
0
votes
0answers
6 views

When is $C(K)$ isomorphic to $C(B_{C(K)^*})$?

Let us think for a moment about the proof the theorem due to Banach and Mazur asserting that $C(\Delta)$ is isometrically universal for all separable Banach spaces (here $\Delta$ is the Cantor set). ...
-1
votes
1answer
27 views

X × Y \ {(x, y)} is path connected if X and Y are both path connected [on hold]

i was solving exercise questions and came across this problem on connectedness ...Let X and Y be path connected spaces and (x, y) ∈ X × Y . If each X and Y has more than one elements then X × Y \ {(x, ...
0
votes
2answers
19 views

tupules $(x,y)$ with at least one entry rational is connected in $R^2$

I have studied connectedness and came across a problem which goes like this.. all the tuples $(x,y)$ with at least one entry rational is connected in $\Bbb R^2$. I have tried to prove it by ...
0
votes
0answers
13 views

Lindelöf subspaces in the product of ordinal spaces [on hold]

Let $$X = W_1 \times ( W_1 + 1 ).$$ How can we describe all Lindelöf subspaces of X ?
16
votes
1answer
544 views

If every open set is a countable union of balls, is the space separable?

Suppose we have a metric space in which every open set is expressible as a countable union of balls. Is this space necessarily separable? Thank you.
0
votes
1answer
27 views

A question on Kronecker Index

I am reading A book by Milnor (Lectures on characteristic classes) and I can across this section on Stiefel-Whitney numbers (page 16) and he uses the Kronecker index but never defines it (He says to ...
2
votes
0answers
23 views

Path Homotopy in a Topological Annulus

Let $C_1$ and $C_2$ be simple, closed curves in $\mathbb{R}^2$ such that $C_1$ lies in the region bounded by $C_2$, and the origin $O$ lies in the region bounded by $C_1$. Define an annulus $A$ as the ...
5
votes
0answers
72 views
+100

On distributivity of lattice of group topologies

Let $\frak L$ be the set of all topologies $\mathcal T$ on $\Bbb Q$ (the additive group of all rational numbers) such that $(\Bbb Q,\mathcal T)$ is a topological group. Then $(\frak L,\subseteq)$ is a ...
1
vote
1answer
50 views

Continuity upon trivial topology

I am puzzled with the following statement: "Given any map $f:X\to Y$ where $X$ is equiped with the trivial topology $(\varnothing,X)$, then this map is continuous iff $Y$ has the trivial topology. ...
0
votes
1answer
38 views

Example of a locally compact metric space whose completion is not locally compact

Can someone suggest an example of a locally compact metric space whose completion is not locally compact?
0
votes
2answers
28 views

Proving that a set $A$ is dense in $M$ iff $A^c$ has empty interior

Prove that a set $A$ is dense in a metric space $(M,d)$ iff $A^c$ has empty interior. Attempt: I think I proved the converse correctly, but I'm not sure how to start the forward direction. ...
1
vote
6answers
2k views

give an example of an infinite class of closed sets whose union is not closed.

give an example of an infinite class of closed sets whose union is not closed. Thanks for your help
0
votes
0answers
9 views

how to do classification of topological space which a poset is a frame

is module in algebraic geometry for classification of topological space which a poset is a frame which invariant is for doing this classification of topological space? if want to do full combination ...
0
votes
0answers
22 views

Computing transition map of $S^2$.

First please have a look at the cruddy diagram I have drawn. (it is at angle because my camera casts a shadow if I photograph it from above) Define the coordinate charts that map a portion of the ...
0
votes
1answer
41 views

How to prove that the Topologist's Sine Curve is a non locally compact space?

Let $R^2$ have the Pythagorian topology. The subspace $Y=\{(0,0)\}\bigcup$ $\{(x,sin(1/x))$ |$ x>0\}$ is usually called the "Topologist's Sine Curve". I wish to prove that Y is a non locally ...
2
votes
2answers
40 views

Counterexamples for $f(\overline{A}) = \overline{f(A)}$ and $\overline{f^{-1}(B)} = f^{-1}(\overline{B})$ in (non-)continuous mapping $f: X \to Y$

Let $f$ be a mapping. Prove that the following three statements are equivalent. $f$ is continuous; $\forall A \subseteq X: f(\overline{A}) \subset \overline{f(A)}$; $\forall B \subseteq ...
4
votes
1answer
66 views

When is $x^{2^n}$ dense in $\mathbb{S}^1$, for $|x|=1$?

Motivation: So I just saw this question: Limit when $n\rightarrow\infty$ of $\text{sgn}(\sin(2^n \pi x))$ with $x\in(0,1)$ fixed., and the answer involves diadic numbers and things of the kind. Most ...
0
votes
0answers
24 views

Point-set topology - derived sets, boundaries, etc.

Consider the metric space $(\mathbb R^2,\sigma)$, where $\sigma$ is the discrete metric. Let $$E=\lbrace(x,y)\in \mathbb R^2 : x^2+y^2<1 \rbrace$$ i.e. the unit disc. Find the following sets ...
2
votes
1answer
85 views

Splitting of Nonmeasurable Sets

Being curious I'm wondering: Let $V$ be a Vitali set defined as usually as a choice of $v\in[r]$ with $0\leq v\leq 1$ for every $[r]\in\mathbb{R}/\mathbb{Q}$. Since the countable disjoint union of ...
0
votes
0answers
21 views

$\{X_\alpha\}_{\alpha\in J}$ family of connected spaces, $X$ a product space. Show $X$ is connected.

Let $\{X_\alpha\}_{\alpha\in J}$ be a family of connected spaces; let $X$ be the product space $$X=\prod_{\alpha\in J} X_\alpha .$$ Let **a**$=(a_\alpha)$ be a fixed point of $X$. (a) Given any ...
0
votes
1answer
26 views

Closure of a certain set

Let $X$ be the ordered square (i.e. $X= [0,1] \times [0,1]$). X is in the order topology. Let $A = \bigl\{\frac{1}{n}\times 1: n \in \mathbb{Z}_+\bigr\}$ a subset of X. Could you please give some ...
0
votes
0answers
16 views

Interior of dense set in ultraregular space

Let $X$ be a regular Hausdorff topological space without isolated point. We say that $X$ is ultraregular if the following condition holds: for every subset $A \subset X$, such that $A$ and ...
5
votes
4answers
261 views

Are these open sets?

This is the question: Which of these sets are open sets on the lower limit topology on $\mathbb{R}$, whose basis elements are $[a,b),a<b$? $$[4,5)\qquad\left\lbrace3\right\rbrace\qquad ...
2
votes
1answer
36 views

Comparing certain topologies

Consider the following topologies on $\mathbb{R}$ $\mathcal{T}_1=$ the standard topology $\mathcal{T}_2=$ the lower limit topology $\mathcal{T}_3=$ the topology having as basis all open rays $(- ...
1
vote
1answer
38 views

Homeomorphism - transforming mug into donut

I read that a map is 'visually' a homeomorphism if you don't have to fold or tear the object. Thus, I was wondering what the problem with folding is? I guess that in this statement they don't assume ...
1
vote
1answer
50 views

Is “connected, simply connected” Redundant?

Here are my definitions of "connected" and "simply connected." A topological space $X$ is connected if and only if it is not the union of two nonempty disjoint open sets. A topological space ...
2
votes
1answer
43 views

Is every closed ball (or open ball) in the Eucledean Space $R^n$ convex?

I am solving a problem and I need to use this fact: Every closed ball (or open) in the Eucledean Space $R^n$ convex? Hoever, I am not sure if it is true or not. Can anyone help? Thanks!
3
votes
2answers
57 views

Is the set of invertible upper triangular matrices open in $GL_n(\mathbb R)$? Is it open in the set of all upper triangular matrices?

I think the answer to the second question is yes, but can't quite prove this. I've no idea about the first part. I've done a few exercises of this kind but all have used the continuity of the ...
0
votes
0answers
12 views

The normalization of a product of varieties

Let $X,Y$ reduced complex analytic spaces, $X^{'}$ and $Y^{'}$ the normalizations of $X$ and $Y$, respectively. Let $(X \times Y)^{'}$ the normalization of $X \times Y$. Is true that $(X \times Y)^{'} ...
0
votes
0answers
10 views

The normalisation map is a bi-Lipschitz map?

Let $X$ a reduced analytic space, $n: W \rightarrow X$ the normalisation map, $W$ the normalisation of $X$ and $S$ the singular set of $X$. When we restrict $n$ to $W\setminus n^{-1}(S)$, we know that ...
1
vote
1answer
33 views

$S^1$ with length metric is not isometric to any subset of Euclidean plane (metric given by restriction)

Let $S^1$ denote point whose radius is 1 from the center. Metric is given by distance between two point is the shortest distance, that is the length metric. Prove that $S^1$ with this metric is not ...
5
votes
5answers
52 views

Is $\{z\in\mathbb C\mid|\text{Re }z|+|\text{Im }z|\le1\}$ open or closed?

I am trying to figure out if the set $\{z\in\mathbb C\mid|\text{Re }z|+|\text{Im }z|\le1\}$ is open or closed or maybe none of that. I hope someone could provide a hint to solve this. Can this set be ...
2
votes
1answer
29 views

Topology making a family of functions optimal

I am trying to do a problem in Arbib's Category Theory book. Loosely rephrased: Let $\{(X_i,\tau_i)\}_I$ be a family of topological spaces, $X$ a set, and $\{f_i:X\to X_i\}_I$ a family of functions. ...
0
votes
1answer
31 views

Searching for analytical or topological proof(s) of the Cayley-Hamilton theorem

Is there any analytical or topological proof(s) of the Cayley-Hamilton theorem ? I want to know such proofs ( if possible ) , I would even appreciate proper references with accessible links . Thanks ...
0
votes
1answer
21 views

Universal Closure Operations

On page 227 in Borceux's Categorical Algebra I, he defines a universal closure operation on a finitely complete category $\mathscr B$ to be an operation on $\overline \square\colon \mathbf{Sub}(B)\to ...
0
votes
1answer
17 views

Question about pointwise convergence of sequences in the box and product topologies.

Can someone please verify my proof or offer suggestions for improvement? I'm aware that there may be answers floating elsewhere, but I need help with my proof in particular. $\textbf{Note:}$ This is ...
2
votes
1answer
303 views

Do all the open sets containing a limit point of an infinite countably compact subset have to contain infinite points?

Say an infinite set is countably compact (if set $E$ is an infinite countably compact set, it contains at least one limit point within itself). Let $x$ be one such limit point of $E$. My textbook says ...
0
votes
2answers
55 views

Are all the points in a nonempty open set limit points?

My conjecture is that given any open set $A\subseteq\mathbb{R}$, all points $a\in A$ are limit points. Prove, or if untrue, disprove by constructing a counterexample. A few definitions for ...
0
votes
1answer
21 views

if $\Bbb B=\{x\in \Bbb R^{n+1}; \langle x,x\rangle<1\}$ be a open ball from Euclidean Space $\Bbb R^n$

I study Metric spaces and I has this problem Show that sphere $\Bbb S^n=\{x\in \Bbb R^{n+1}; \langle x,x\rangle=1\}$ is metrically homogeneous. For the other hands, if $\Bbb B=\{x\in \Bbb R^{n+1}; ...
21
votes
3answers
652 views

Are there uncountably many non homeomorphic ways to topologize a countably infinite set?

Today I was fooling around a bit trying to count the topologies on a finite set. I didn't make much progress, so I did some googling and quickly discovered it is an open problem to give a closed form ...
1
vote
0answers
31 views

Is there a model of set theory in which $2^{2^{\omega_1}}$ is separable?

We know that $2^{\mathfrak c^+}$ ($\mathfrak c =2^\omega=|\mathcal P (\omega)|$) is not separable by the following argument: Suppose $D$ is countable dense in $2^{\mathfrak c^+}$. For each ...
0
votes
2answers
36 views

For a compact set $K\subset \Bbb R^n $ prove the following :

For a compact set $K\subset \Bbb R^n $ and $\delta>0$ show that that there exists a finite number of elements in $K$, say $x_1,x_2,\dots,x_k$ such that any other element $x$ of $K$ is at a ...
0
votes
1answer
29 views

Problem with topological space in probability theory. [on hold]

Let $(X, \tau)$ be a topological space. a) Show that arbitrary intersections of closed sets are closed. b) Prove that a set $F \subseteq X$ is closed if and only if for all sequences $\{x_{n}\} ...
0
votes
0answers
39 views

Prove that $F$ is continuous iff for each $i$ $F_i$ is continuous.

let $(X,\tau)$ and $(Y_i,\sigma_i)\space i\in \mathbb N$ be topological spaces. let $Y:=\prod_{i=1}^\infty Y_i$ and $\sigma:=\prod_{i=1}^\infty \sigma_i$ the product topology (for any $U_1\times ...
0
votes
2answers
56 views

Let $ A\subset B$ be a closed and bounded set, and let $\sup(A)=b$. Show that $b \in A$.

Let $ A\subset R$ be a closed and bounded set, and let $\sup(A)=b$. Show that $b \in A$. I understand the concept but not quite sure where to begin for the proof.
2
votes
2answers
29 views

prove that any continuous bijection from $S^2$ to itself is is an homeomorphism

Let $S^2=\{(x,y,z)\space:\space x^2+y^2+z^2=1 \}$ be a subspace of $(\mathbb R^3,d_{euclid})$. prove that every continuous bijection $F\space:\space S^2\rightarrow S^2$ is an homeomorphism from $S^2$ ...
5
votes
0answers
45 views

How to master general topology for analysis?

I started learning topology long ago. I first exposed myself to metric topology in Baby Rudin and Munkres Topology 2nd ed. Part I. Munkres is my most revisited book ever since. The first big ...
0
votes
1answer
13 views

Proving convergence iff projection converges in product topology

This question is regarding the same problem. I wish to present my proposed solution and get feedback on my argument, and as such, I claim that it is not a duplicate. (In particular, the other asker ...