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 (3)

1
vote
0answers
23 views

Prove $ℝ^I$ is not Lindelöf

Munkres's topology book has a large review exercise, which includes testing topological axioms on a variety of spaces. One of those spaces is $ℝ^I$ where $I=[0,1]$ is the unit interval, and one of the ...
4
votes
1answer
69 views

If $f(\mathbb{R})$ is compact and $f$ is continuous, then is $f$ uniformly continuous?

Question: If $f(\mathbb{R})$ is compact and $f$ is continuous, then is $f$ uniformly continuous? Background: I thought of the question when proving that "If a function is periodic and continuous, ...
3
votes
3answers
47 views

Is every countable ordinal homeomorphic to a subspace of $\mathbb R$?

I know that every countable ordinal is isomorphic to some subset of $\mathbb R$ as ordered sets. Is it also the case that every countable ordinal (with the order topology) is homeomorphic to some ...
0
votes
0answers
29 views

Proving least upper bound property implies greatest lower bound property

In Rudin 1.11 Theorem Proof he claims the following Suppose $S$ is an ordered set with the least upper bound property $B \subset S$, $B$ is not empty, and $B$ is bounded below. Let $L$ be the set of ...
0
votes
0answers
22 views

Compact open topology and nets

If $X$ and $Y$ are topological spaces we can form a topology on $Y^X$, which has as subbasis sets of the form $B(T,U) := \{f \in Y^X : f(T) \subset U \}$ where $T$ compact and $U$ open. Is there a ...
-1
votes
1answer
71 views

Is $d(x,y)=|x-y|^2$ a distance on $\mathbb{R}$?

Please how to prove that $d(x,y)=|x-y|^2$ is a distance on $\mathbb{R}$, I don't know how to solve the triangular inequality. Thank you.
1
vote
3answers
50 views

Topology; Definition of the open ball and open sets confuses me

I just picked up T.W Gamelin’s book on topology. I started reading and got confused when I came to the definition of an open ball on the second page. It says $B(x;r) =$ All $y$ in the set $X$ such ...
4
votes
2answers
270 views

Proving an intuitive fact about sets in the plane

The entire 2-dimensional plane is covered by 3 sets: Blue, Green and Red. It is given that: All sets are closed. All sets are interior-disjoint (but may meet at their boundaries). Blue is bounded. ...
0
votes
4answers
53 views

If a continuous function satisfies $|f(z)^2-1|<1$ for every $z$, then either $|f(z)-1|<1$ of $|f(z)+1|<1$ for every $z$

Suppose a continuous function $f:D\rightarrow \mathbb{C}$ where $D$ is a plane domain, has the property $|f(z)^2-1|<1$ for every $z$ in $D$. Show that $|f(z)-1|<1$ of $|f(z)+1|<1$ for every ...
1
vote
1answer
25 views

Topological Spaces: Pre-Uniform Structures

Disclaimer This thread is meant to record. See: Answer own Question Reference It is a follow-up to: Uniform Spaces: Neighborhood System It has relevance to: TVS: Uniform Structure Problem Given ...
0
votes
1answer
15 views

Conditions so that Lebesgue Covering Dimension and “Usual” Dimension are Equal

The definition of covering dimension is as follows: The ply of a cover is the smallest number $n$ (if it exists) such that each point of the space belongs to at most n sets in the cover. A refinement ...
1
vote
1answer
40 views

Using arctan to prove equivalence of 3 definitions of a manifold

My uncle is mathematician and a bit of a wise guy. He challenged me to use the properties of arctan to answer a particular problem but I have no idea what I'm looking for or why arctan is a good ...
2
votes
1answer
30 views

Subbases and half-planes

If $(X,d)$ is a metric space, it's easy to show that $H(x,y)=\{w\in X\mid d(x,w)>d(y,w)\}$ is open in the topology $\tau$ induced by $d$. Is, in general, $\{H(x,y)\mid (x,y)\in X\times X, x\neq ...
2
votes
0answers
33 views

Two disjoint connected and bounded open sets in the plane that shares the same boundary

In $\mathbb{R}^2$ with std. topology I want to exhibit two open sets that are connected, bounded and disjoint but that have a common boundary. My attempt: Since both my sets need to be bounded, my ...
-1
votes
0answers
54 views

How to prove that an injection from a sphere into a Euclidean space is homotopic to a constant?

How to prove that the injection $i: S^{m-1}\rightarrow \mathbb{R}^m$ is homotopic to a constant ? Where $S^{m-1}=\{x\in \mathbb{R}^m, |x|=1\}$ Thank you.
1
vote
1answer
95 views

Questions about a topological category

Given topological spaces $(X_i,\tau_i)$ with sets $\mathbf S_i=\{\mathcal A\in \tau_i^2|\mathcal A\supseteq\Delta X_i^2\}$, where $\tau^2$ is the product topology and $\Delta$ is the diagonal. The ...
0
votes
1answer
60 views

Dimensions definition?

Apparently dimension is "informally" defined as "the minimum number of coordinates needed to specify any point within it". For example we need at least 3 numbers to describe any point in the 3D space ...
3
votes
2answers
34 views

Regular spaces, weight and dense subsets

It is known that in case a space $X$ is regular ($T_1$ + $T_3$), its weight is less than or equal to $2^{d(X)}$, where $d(X)$ is the density of $X$. What is an example of a Hausdorff space for which ...
7
votes
1answer
69 views

Topology and Measures

I apologize if this question is a bit vague; I'm just wondering if there is a concept like what I'm talking about, or if I'm just lost. I'll start with just some thoughts. I looked a bit, and I don't ...
3
votes
0answers
43 views

Show that a map of sets is continuous if its composition with other functions is

Problem: Let $Y, E, B$ be topological spaces with $Y$ locally path connected. Suppose $p: E \rightarrow B$ is a covering map, with $g: Y \rightarrow E$ a map of sets. If $p \circ g$ is continuous, ...
5
votes
1answer
74 views

Density of the rationals in the reals

While studying measure theory I have encountered the following set, $$U_\varepsilon=\bigcup_{n\in \mathbb{N}}(q_n-\varepsilon /2^n,q_n+\varepsilon/2^n),$$ where $(q_n)_{n\in \mathbb{N}}$ is an ...
4
votes
1answer
47 views

When does a continuous function defined on a non-compact closed and bounded convex set has a fixed point?

Is there any result in fixed point theory which will give the existence of a fixed point for a continuous function defined on a non-compact, closed and bounded convex set?
6
votes
1answer
75 views

inverse limit in the plane

What stuff can I say about inverse limits regarding the mapping of $[0, 1]$ onto $[0,1]$ given by $$f(x) = \left\{ \begin{array}{ll} 2x & \mbox{if } 0 \le x \le {1\over2}\\ 1 & \mbox{if ...
2
votes
1answer
56 views

elementary topology exercises reference

Can anyone recommend a good collection of elementary topology exercises? A pdf collection of undergraduate problem sets and homework, or midterm and final exams that I could practice on? Even a ...
3
votes
2answers
39 views

Directed sets to describe a topology with nets.

I'm studying some things related to ultrafilters on metric and topological spaces and trying to prove theorem in a general setting, so the following question came to my mind. Let $S$ be a ...
0
votes
1answer
22 views

Baire property for finite discrete spaces

Does it makes sense to assume that a nonempty open set of a finite discrete topological space has the Baire property?
0
votes
0answers
34 views

Bounded polyhedrons

Given a bounded polyhedron $P=P(A,b)$ and with $x$ s.t. $Ax<b$, show: $\exists \ \alpha>0 \ \ \ \ \ \text{ s.t.}\ \ \ \alpha^Tx\leq1, \ \ \ \ \forall x \in P $ How I should proceed to prove ...
0
votes
2answers
40 views

queston about T4- spaces

please can can anyone tell me that is it is true? i found this statement (question) in one old book of topology. i think it is just printing mistake. statement(question) is prove that every T4 space ...
3
votes
2answers
47 views

Simply connected and connected in complex analysis

Can some one please help me with this, why is third set in the picture not simply connected. The definition of simply connected (in space of complex numbers) is: A set is said to be simply ...
1
vote
3answers
36 views

Is ${\mathbb R}^n$ with the product topology the same as the metric topology

I have looked at several places into the definition for product spaces. Now all of the definitions I have seen, define the product space topology as generated from the product of sets $U_i$, for which ...
1
vote
1answer
32 views

A question about a perfect space and a linear order on it

Suppose I have a nonempty perfect Polish space $X$, and there's some linear order $<$ on it (it is not related to the topology on $X$ in any way). How can I prove that there is a point $y$ in $X$ ...
2
votes
4answers
61 views

Topological Spaces Involving Connectedness, Compactness, and Hausdorfness

I made a table about topological spaces with or without connected, compact, and Hausdorff properties. However, I cannot find the example for the following cases: Compact, but neither Hausdorff nor ...
2
votes
1answer
34 views

Covering map is proper $\iff$ it is finite-sheeted

Prove that a Covering map is proper if and only if it is finite-sheeted. First suppose the covering map $q:E\to X$ is proper, i.e. the preimage of any compact subset of $X$ is again compact. Let ...
0
votes
1answer
28 views

about finite discrete space

question : is all finite discrete spaces are $T_2$- space, $T_1$ -space and also $T_0$ -space. i have taken very simple example: X = {a, b, c} and topology $T = \{ ∅, X, \{a\}, \{b\}, \{c\}, ...
2
votes
0answers
50 views

Dual of path in a space.

Is there a notion dual to the notion of a path in a topological space? Given that a path in a space $X$ is a continuous function from the interval $[0, 1]$ to X, what would the dual of this notion be, ...
0
votes
0answers
28 views

software for drawing sequence in metric space [on hold]

Use what kind of software to draw a grapf describing open cover and a sequence in a metric space? For example, I need to show some subsets of an open cover and a sequence consists of many points in a ...
0
votes
1answer
23 views

Question concerning one rule of the topological calculus in terms of the interior operator

Define a topological space as a set $X$ and a function $\text{int}()$ assigning to every set $A\subseteq X$ the set $\text{int}(A)\subseteq X$ such that: (i) $\text{int}(A\cap ...
0
votes
1answer
30 views

Zero sets in completely regular spaces

I'm wondering if I am missing something from this portion of a problem (14.C.1 - Willard) A zero set in a topological space $X$ is a set of the form $f^{-1}(0)$ for some continuous ...
3
votes
2answers
89 views

$[0,1]\times[0,1]$ stays connected after removal of an interior point

I am self-studying Topology's connectedness and came across this simple problem: Show that $[0, 1] \times [0, 1]$ stays connected if you remove an interior point. Visually it looks simple ...
0
votes
2answers
61 views

How to find a continuous function that demonstrates that the set $\{(x,y):y>x\}$ is open?

Consider the set of points $U$ in $\Bbb{R}^2$ that lie above the line $y = x$, i.e. points $(a,b)$ such that $b>a$. Prove that $U$ is open and connected. The method that is recommended is showing ...
-1
votes
0answers
21 views

Convex basis and conical basis (how to draw?)

There is a question, I'm struggling with: Find vertices of the following described polyhedron, $P:=P(A,b)=conv(V)+cone(E)$ where $V$ is the set of all vertices of $P$ and $E$ is the set of all ...
0
votes
2answers
33 views

Proof check: proving a neighborhood is an open set?

I want to prove that a neighborhood is an open set by picking an arbitrary point in it and showing it's an interior point. On my final exam I couldn't think of a way to use the triangle ...
1
vote
0answers
34 views

$E$ is compact if and only if X is compact and $q$ is a finite-sheeted covering

Let $q:E\to X$ be a covering map. Then $E$ is compact if and only if X is compact and $q$ is a finite-sheeted covering. My question is regarding the $"\implies"$ direction: If $E$ is compact, then ...
0
votes
1answer
30 views

Tychonoff spaces with small weight

Let $\kappa$ be an infinite cardinal. Is there a Tychonoff space $(X,\tau)$ such that $|X| = 2^\kappa$ and $(X,\tau)$ has weight $\kappa$ (= a basis consisting of $\kappa$ elements)?
4
votes
2answers
45 views

Two-sheeted covering of the Klein bottle by the torus

Prove that there is a two-sheeted covering of the Klein bottle by the torus. OK, so we take the the polygonal representation of the torus and draw a line in the middle as follows: Then there are ...
2
votes
1answer
38 views

If $X$ is Hausdorff, then so is $E$

Let $q:E \to X$ be a covering map. If $X$ is Hausdorff, then so is $E$. OK, suppose $X$ is Hausdorff and let $x,y \in E$ with $x\neq y$. Let $V$ denote the evenly covered neighbourhood for $q(x)$, ...
6
votes
1answer
105 views

Show that $\{e^{in}: n\in\Bbb N\}$ is Dense in the Unit Circle

This problem gave me fits when I was in grad school. Looking back at it now, it still escapes me. The problem is from Conway's Functions of One Complex Variable. I'm looking for a proof from basic ...
3
votes
1answer
34 views

Discrete subspace of $\mathbb{N}^\mathbb{N}$

Endow $\mathbb{N}$ with the discrete topology. Does $\mathbb{N}^\mathbb{N}$ contain a discrete subspace of size $2^{\aleph_0}$?
1
vote
3answers
51 views

Is it possible to get a neighborhood with only finitely many points in it, in an infinite set?

If we have an infinite set, is it possible to find a neighborhood around a certain point in the set that has only finitely many points in it?
0
votes
0answers
38 views

How do I show $G_0$ and $G_1$ are conjugate subgroups? Please improve my answer.

Is my solution below correct? Please read through it and tell me if it seems complete or to make sense. Question: Let $E$ be path-connected. Let $p : E → B$ be a covering map and $p_∗$ be the ...