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)

1
vote
1answer
29 views

What sequences could satisfy these requirements?

I need to find a sequence which converges to $0$ but is not in any space $\ell^p$, where $1 \leq p < +\infty$. And, I need to find a sequence which is in every space $\ell^p$ with $p > 1$ but ...
0
votes
0answers
40 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.
0
votes
1answer
32 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 ...
1
vote
0answers
26 views

Continuous maps vs. open maps [duplicate]

When we study topology, we typically study topological spaces and continuous maps between them. From a categorical perspective, this is "wrong," because continuous maps are not the structure ...
2
votes
1answer
30 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. ...
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 ...
1
vote
1answer
30 views

Where was I supposed to use compactness in this proof that a compact subspace can be separated from a point by open sets?

Can anyone check my proof below? P. Let $X$ be a metric space. Prove that if $K\subseteq X$ is compact and $x\notin K$, there exist disjoint open sets $U$ and $V$ such that $K\subseteq U$ and ...
2
votes
1answer
58 views

How to conceptualize unintuitive topology?

I found Project Origami: Activities for Exploring Mathematics in my university's library the other day and quickly FUBAR'd (folded-up beyond all recognition) the couple sheets of paper I had with me ...
3
votes
0answers
12 views

Show that if $X_\alpha$ is Hausdorff for all $\alpha$, then $\prod X_\alpha$ is Hausdorff under the box and product topologies.

Can someone please verify my proof? I am aware that there is a similar question posted elsewhere, but I need help with my proof in particular. $\textbf{Note:}$ This is not homework. Show that if ...
0
votes
2answers
57 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
0answers
19 views

Properties of normal set

For any closed subset $F$ of $X\subset \mathbb{R}^{n}$, we define the normal set $\mathcal{N}(F)$of $F$ as follows: if there exists $f\in C^2(X)$, and $x_0\in F$, such that $$ df(x_0)\neq 0;\\ ...
0
votes
2answers
38 views

Determine if the following set is open, closed or neither {$x\in\mathbb{R}^{2} | x_1 + x_2 = 1$} $\subset \mathbb{R}^{2}$

my textbook doesn't have solutions to most these problems, and this one is really giving me some trouble. Any help is appreciated. Determine if the following set is open, closed or neither ...
-1
votes
0answers
25 views

$K$- Topology .aaa [closed]

Define the $K$-topology of $\mathbb R$ as follows: Let $K=\{1/n | n\in \mathbb N\}$ and let $B_k$ be the collection of the subsets of the form $(a,b)-K$ in $\mathbb R$ and all open intervals. The ...
1
vote
1answer
34 views

Is the Riemann sphere homeomorphic to $\overline{\mathbb{R}}\times\overline{\mathbb{R}}$?

Let $\hat{\mathbb{C}}$ be the Riemann sphere. Let $\overline{\mathbb{R}}$ be the extended real. (i.e. $\mathbb{R}\cup\{\infty,-\infty\}$) Then, is $\hat{\mathbb{C}}\cong ...
0
votes
2answers
35 views

Why is this set connected?

I don't understand something about a proof of that if $X_i$ is a connected space for every $i\in I$, then $X=\Pi_{i\in I}X_i$ is connected. It is this: Let $x\in X$. Define $C$ the set of all $y\in ...
2
votes
1answer
28 views

Doubts about definition of open sets in “Understanding Analysis” by Stephen Abbott

In the book "Understanding Analysis" by Stephen Abbott, the author defines an open set as: A set $O \subseteq \mathbb{R}$ is open if for all points $a \in O$ there exists an ...
2
votes
0answers
35 views

How many Y shapes can you fit on the plane?

You can only fit at most countably many disjoint open discs on the plane: for any collection of disjoint open discs, it is possible to pick a single rational coordinate contained in each disc, and ...
0
votes
1answer
30 views

For $f:D\subset \Bbb R^n \rightarrow \Bbb R^m$ prove the following are equivalent:

For $f:D\subset \Bbb R^n \rightarrow \Bbb R^m$ prove the following are equivalent: a)$f$ is continuous in $D$ b)If $O\subset \Bbb R^m$$f$ is an open set, then there exists an open set $G\subset ...
1
vote
1answer
57 views

An open set in $\mathbb{R^n}$ is connected if and only if it is path connected

Here is a proof I found on the internet but cannot understand a part of it which is highlighted. I hope someone can help me understand this. Thanks in advance
6
votes
0answers
109 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 ...
0
votes
1answer
24 views

Is there any special name for a $n$-torus made by products of hyperspheres?

I was wondering if there exist an accepted name for an $n$-torus made by the product of hyperspheres $\mathbb{S}^d$, that is for the following set: $$ ...
3
votes
1answer
55 views

When do functions turn a space into a locally ringed space?

Let $X$ be a topological space, and consider for each open set $U \subseteq X$ a set $F_U$ of functions $U \to k$ into some fixed field $k$. Let $\mathcal{O}$ be the sheaf of $k$-algebras induced by ...
1
vote
2answers
37 views

How to prove that $H(S_1\cap S_2)\subset H(S_1) \cap H(S_2)$ and $H(S_1 \cup S_2) \supset H(S_1) \cup H(S_2)$

I'm studying convex analysis and my task is to prove the following inclusions: $S_1, S_2$ are non-empty sets in $\mathbb{R}^n$, and $H(S) $ defined as the convex hull of set $S$. Show that ...
1
vote
0answers
32 views

Do two data sets have the same distributions?

$X = \{x_i\}_{i=1}^n \in \mathbb{R}^d$ is a data set, Whether there exists another data set $Y=\{y_i\}_{i=1}^n \in \mathbb{R}^m (d>m)$ so that the distribution of X and Y are approximately ...
0
votes
1answer
30 views

What does it mean “sequence with infinite range”

I'm trying to understand this phrase Find a sequence with infinite range that converges only to $0$. What does it mean "sequence with infinite range"? Thanks
0
votes
0answers
24 views

Show that if $F$ is continuous, then it is continuous in each variable separately.

Can someone please verify my proof? I am aware that there may be a similar question posted elsewhere, but I need help with my proof in particular. $\textbf{Note:}$ This is not homework. Let $F: X ...
1
vote
3answers
67 views

Proof of topological isomorphism

I remember reading in a section in plato.stanford.edu that the interval $(-∞, t)$ is topologically isomorphic to the interval $(0, t)$. I am not that good with topology, so could someone show me the ...
7
votes
3answers
130 views

Why are continuous functions the “right” morphisms between topological spaces?

Recently, someone mentioned to me that given a function $f: X \to Y$ there are two natural functions between the powersets $P(X)$ and $P(Y)$. Namely $f: U \subset X \mapsto f(U)$ and $f^{-1}: V ...
0
votes
1answer
28 views

Finding simple homotopy type

I have an excercise that I kind of dislike: Given $T-\{p,q\}$ where $T = S^1 \times S^1 $ and $p,q \in T$ two different points, I am supposed to find a simple homotopy equivalent space by ...
0
votes
0answers
40 views

Transitivity of smooth submanifolds

I was reading through Guillemin and Pollack and was having trouble verifying this for myself. Given $M \subset N$ and $N \subset P$, where $M$ is a submanifold of $N$, and $N$ a submanifold of $P$, ...
0
votes
0answers
20 views

Define the 4 types of interval subsets of the real numbers.

Define the 4 types of interval subsets of the real numbers. Is the union of an arbitrary number of open intervals also an open interval? Is the intersection of an arbitrary number of open intervals ...
0
votes
1answer
56 views

Example of sigma algebra that is not a topology

There is a very nice explanation of an example of sigma algebra that is not a topology: here. I do not fully understand the answer. Apparently this is a basic question, but why do we want this ...
0
votes
0answers
20 views

Is there a reason why M can't be all summable sequence?

Let M be the set of all summable non-negative sequences $\{x_k\}_{k=1}^\infty$ of real numbers, that is, $x_k \geq 0$ for all k and $\sum_{k=1}^\infty x_k$ converges to a real number. Let $d:M \to ...
4
votes
1answer
43 views

Can measure induce a topology on a Set?

When I was taught metric spaces in Topology, I came across the idea that metric defined on a set can induce a topology by creating a basis (open balls). If we have a measure defined on a set, can it ...
2
votes
2answers
51 views

What is $d(\sin(x),\cos(x))$ if d is a distance function in a metric space?

Let $M=\{f:[a,b] \to \textbf{R} | f \,is \,continuous \}$. Let $d:M \to \textbf{R}$ be defined by $d(f,g)=\int_a^b |f(x)-g(x)| \,dx$. What is d represent geometrically, and show that M, d is a metric ...
4
votes
0answers
63 views

Equivalence of Lebesgue Measurablity

Hello Mathematics Community. I am having some difficulties with the following problem dealing with Lebesgue Measure and its equivalent interpretation. I will first include the definitions which I am ...
2
votes
2answers
80 views

Why does the product topology allow proper subsets for only finitely many elements?

Consider Theorem 19.1 from Munkres' topology: The box topology on $\prod X_\alpha$ has as basis all sets of the form $\prod U_\alpha$, where $U_\alpha$ is open in $X_\alpha$ for each $\alpha$. The ...
0
votes
0answers
60 views

How to organize my learning in Maths?

I m working on a problem in mechanics of material which concerns about the variation of shapes. I need to understand the deformation of material. I m a civil engineering graduate. All my understanding ...
0
votes
2answers
84 views

Constructing a circle from a square [duplicate]

I have seen a [picture like this] several times: featuring a "troll proof" that $\pi=4$. Obviously the construction does not yield a circle, starting from a square, but how to rigorously and ...
1
vote
1answer
39 views

Two definitions of connectedness: are they equivalent?

A topological space $(X, \tau)$ is connected if $X$ is not the union of two nonempty, open, disjoint sets. A subset $Y \subseteq X$ is connected if it is connected in the subspace topology. In ...
1
vote
1answer
44 views

How Can I prove the three statements are equivalent?

Let $X$ be a compact Hausdorff space and $f:X \rightarrow Y$ be a quotient map. Show that the following are equivalent: (a)$Y$ is an Hausdorff space, (b)$f$ is closed map, (c)The set ...
0
votes
0answers
24 views

Finding the identification topology

I'm reading the book of Dugundji under Identification topology, and as stated: Let $p: I \rightarrow {\{0\}\cup \{1\}}$ be the characteristic function on $[1/2,1]$. Then the mapping should i think ...
1
vote
0answers
40 views

Point-set topology, characterizing sets

I think the question sort of speaks for itself. Please no solutions in the answers - I'm mostly looking to see if my logic makes sense: Let $E$ be a set in a metric space $(X,d)$. Let $E^{\circ}$ ...
0
votes
1answer
37 views

Classification of proper maps in topological spaces

How can I prove that if $f:X \to Y$ is continuous of locally compact, Hausdorff topological spaces, then $f$ is proper (inverses of compact sets are compact) iff it extends continuously as a map ...
0
votes
2answers
60 views

Determine if $\mathbb{R}$ \ $\mathbb{N}$ is open closed or neither

Determine if $\mathbb{R}$ \ $\mathbb{N}$ is open closed or neither. I've been on this problem for a while now. As of right now I pretty confident its neither because I dont really see how it can be ...
1
vote
1answer
31 views

Isolated points are open if $|x| \geq 2$?

Theorem: Define $X$ to be a topological space with $|X| \geq 2. $ Then $x \in X$ is an isolated point$\iff$ $\{ x \}$ is open. I am reading this and the proof proceeds with a neighbourhood $U$ of ...
1
vote
1answer
27 views

The closed set in the product topology

We know that for the product topology $X\times Y$, the open sets are generated by $U\times V$,where $U,V$ are open in $X,Y$ respectively. I am considering the closed sets in $X\times Y$, are they ...
3
votes
0answers
56 views

Compactness and Lipschitz functions

I am very stumped by this question: Suppose (K, d) is a compact metric space. Let f be any function, f: K $\rightarrow \mathbb{C}$, not necessarily continuous. Prove that for any $\epsilon > ...
1
vote
0answers
28 views

Showing a subset of $\;\Bbb R^2\;$ cannot be the set of limit points of any other set

I will appreciate any insight in the following proof (if, indeed, it is a proof): Let $$F:=\left\{\;(x,y)\in\Bbb R^2\;;\;\;xy\in\Bbb Q\;\right\}$$ Prove that there doesn't exist $\;A\subset\Bbb ...