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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2
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1answer
51 views

How to prove the limit of Thomae function?

Given the Thomae Function: $$t(x)= \begin{cases} 1 & \text{if }x=0, \\ 1/n & \text{if }x=m/n \in \mathbf Q\setminus\{0\}\text{ is in lowest terms with }n>0,\\\ 0 & ...
0
votes
0answers
24 views

Coverings maps of a simply connected space

Let be $Y$ a simply connected space. Show that $Y$ doesn't admit covering maps that aren't homeomorphisms, ie, every cover space of $Y$ is trivial ($I\times Y$, with $I$ a discrete space). So, I know ...
1
vote
1answer
30 views

Proving continuity on a generated topology

Say we have a space $X$ and a collection of functions $\{f_i:X \to Y\}$, where $Y$ is endowed with some topology $\mathcal{T}_Y$, and $\mathcal{T}$ is the smallest topology making all $f_i$ ...
1
vote
2answers
20 views

Condition that a local homeomorphism be a covering map.

Let be $f:Y\to X$ a local homeomorphism, with $Y$ a compact space and $X$ a Hausdorff connected space. How can I show that, for each $x\in X$, $p^{-1}(x)\subset Y$ is finite? So, is clear that $f$ is ...
-2
votes
1answer
54 views

Mayer-Vietoris sequence [closed]

How do I compute the homology of the space obtained by taking three copies of $D^n$ and identifying their boundaries with each other?
1
vote
1answer
56 views

Is it possible for an uncountable subset $S$ of $\Bbb R$ to satisfy $∂S = S$?

Is it possible for an uncountable subset $S$ of $\Bbb R$ to satisfy $∂S = S$? Please shed some light to it. I am not getting any clue to it.
1
vote
0answers
72 views

Prove that if $T$ is one-to-one on $D$, then the set $T(D)$ is open

Let $f$ and $g$ have continuous first-order partial derivatives on an open set $ D\subseteq\mathbf{R}^2 $ and let $T :D \to \mathbf{R}^2 $ be defined by $ T(u,v)=(f(u,v),g(u,v)). $ ...
3
votes
1answer
21 views

A metric space in which no non-empty countable subset has empty interior.

A Question: Find a metric space in which no non-empty countable subset has empty interior. My Answer: In a discrete metric space $X$, $int(S) = S$ for all $S ⊂ X$. And if $S$ is non-empty (and ...
0
votes
3answers
43 views

Is a point a neighbourhood?

In this question Can a neighbourhood of a point be an singleton set? The answers indicate that a point is a neigbourhood. But would this therefore not mean that every (continuous) function is ...
0
votes
1answer
59 views

Which of these are homotopy equivalent? $S^1, \mathbb{R}, \{*\}$

Which of these spaces are homotopy equivalent: $S^1, \mathbb{R}, \{*\}$? I found a homotopy equivalence between $\mathbb{R}$ and the one point space $\{*\}$, so they are homotopy equivalent. The ...
1
vote
2answers
44 views

Proving that a continuous map is homotopic to the constant map

How can I prove that a continuous map $f : \mathbb{R}P^2 \to S^1$ is homotopic to the constant map? I know that in the projective space every point is a line but I do not get why the above has to be ...
-1
votes
1answer
34 views

Topology question. [closed]

If $A$ is a subset of $\mathbb{R}$ , let $a\in A$ and $A\setminus\left\{a\right\}$ is compact.then which of the following are true:- $A$ is compact, $A$ is connected, $A$ is finite set, $A$ is ...
1
vote
1answer
43 views

cohomology ring of a quotient space

what is the cohomology ring $$ H^*((S^3\times S^3\setminus \{e\})/(a,b)\sim (ab,b^{-1});\mathbb{Z}_2)? $$ Here the unit $e$ and the product $ab$ is of the Lie group $S^3=Sp(1)$.
0
votes
2answers
35 views

A question about topologies on the product of two topological spaces.

I am trying to show that if (X, Y) are two topological spaces, then $X\times Y$ with the product topology is the coarsest topology for which the projection functions on $X$ and $Y$ are continuous. I ...
1
vote
1answer
30 views

cohomology ring of some spaces [closed]

What is the cohomology ring $$ H^*(\mathbb{R}^2\times S^1\setminus (0,1);\mathbb{Z})? $$ $$ H^*(\mathbb{R}^2\times S^3\setminus (0,e);\mathbb{Z})? $$ Here $0=(0,0)\in \mathbb{R}^2$, $1=\exp(i0)\in ...
2
votes
1answer
63 views

Question about the wording of a topology problem.

I was asked to show that the topology $\mathcal{T}_{X\times Y}$ is the smallest topology for which the functions $$f_X:X\times Y \rightarrow Y , f_X((x,y))=x $$ and $f_y$ are continuous (where $f_Y$ ...
1
vote
1answer
53 views

Mapping torus with homotopic homeomorphisms

Suppose I define the mapping torus $M_f$ in the usual way by identifying $(x, 0)$ and $(f(x), 1)$. If I have a homeomorphism $f: X \rightarrow X$ and another homeomorphism $f': X \rightarrow X$ that ...
4
votes
1answer
21 views

A meagre set is always contained in an $F_σ$ set made from nowhere dense sets.

In this page I have found a beautiful result that a meagre set need not be an $F_σ$ set (countable union of closed sets), but is always contained in an $F_σ$ set made from nowhere dense sets. Also ...
0
votes
0answers
30 views

Properties of a metric

Consider the set $X=C([0,1])$. Define the function as $$\rho(f,g)= \int_{0}^{1}{x|f(x)-g(x)|dx}$$ $f,g \in X$. a) Show that $\rho$ is a metric in X. b) If $\lambda \neq 0$, then $$\lambda ...
2
votes
1answer
19 views

How to show that boundary of unit ball is empty in ultrametric spaces

Let $(S,d)$ be an ultrametric space. According to wikipedia, any ball must have empty boundary. Why is this true? I am unable to prove this.
-2
votes
1answer
54 views

Continuous Functions and characteristic function

Let $X$ be a topological space and let $\chi_A : X \rightarrow \mathbb{R}$ be the characteristic function for some subset $A$ of $X$. Show that $\chi_A$ is continuous at $p\in X$, if only if $p$ is ...
0
votes
1answer
40 views

Uniqueness of the limit of a function for Hausdorff spaces that are not metric spaces

Is there a proof to show that a mapping from an arbitrary topological space to an arbitrary Hausdorff space (excluding metric spaces) has a unique limit point? Therefore, the only properties to be ...
-1
votes
0answers
42 views

Topologies induced by functions [on hold]

Show that the coarsest topology on the real line $\mathbb{R}$ with respect to which the linear functions $f:\mathbb{R}\rightarrow (\mathbb{R}, U)$ defined by $f(x)=ax+b$, $a,b\in\mathbb{R}$ are ...
11
votes
3answers
131 views

Existence of continuous angle function $\theta:S^1\to\mathbb{R}$

Let $S^1\subseteq\mathbb{C}$ be the unit circle and let $U\subseteq S^1$ be open. How to show that there exist a continuous function $$\theta:U\to\mathbb{R}$$ such that $$e^{i\theta(z)}=z$$ for all ...
3
votes
2answers
23 views

Compact subset of space of matrices and compactness verification of a set of eigenvalues

Let $M_n(\mathbb R)$ be the vector space of real matrices of size $n$ , identified with $\mathbb R^{n^2}$ ; let $X \subseteq M_n( \mathbb R)$ be a compact set ; let $S \subseteq \mathbb C$ be the set ...
1
vote
1answer
33 views

Name/Topological properties of the space of formal power series $\mathcal K [x]$

So, a guest lecturer introduced a concept the other day in class. Take a field $\mathcal K$ and then take the ring of formal power series on that ring, $\mathcal K[x]$. Ignoring convergence in the ...
1
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0answers
33 views

Topology of $L^2$ space

Cardinality of space of all funcions $f: \mathbb R \rightarrow \mathbb R$ is $\beth_2$. However, cardinality of space of all such square-integrable functions, space $L^2$, is $\beth_1=\mathfrak c$, ...
1
vote
1answer
25 views

Proving $g(y)f^{''}(x)+f(x)g^{''}(y)+ (\alpha ^2 + \beta ^2)f(x)g(y)= 0 \implies f^{''}(x)+\alpha ^2f(x) = 0 , g^{''}(y)+ \beta ^2 g(y)=0$

I was reading a proof in physics, suddenly I'm stuck at proving this passage: $g(y)f^{''}(x)+f(x)g^{''}(y)+ (\alpha ^2 + \beta ^2)f(x)g(y)= 0 \implies f^{''}(x)+\alpha ^2f(x) = 0 , g^{''}(y)+ \beta ...
2
votes
1answer
40 views

Questions about sigma-algebra

I am learning measure theory this semester. The definition for sigma-algebra is "a collection of sets that is closed under complements and countable unions and intersections." I wonder what does it ...
5
votes
1answer
29 views

Two geometrical objects in same dimensional plane are homeomorphic.

What can be a good way to prove that two geometrical objects in same dimensional plane are homeomorphic?? For example....to show that a circle and a ellipse is homeomorphic in $\Bbb R^2$ and a ...
0
votes
2answers
51 views

Show that a set is connected and contractible?

I am having trouble with a practice qual exam question: Let $X = \{x,y\}$ with $\emptyset,X,\{x\}$ open. Show that X is connected and contractible? For the first part, I would assume not. That ...
0
votes
3answers
48 views

Product of Two Metrizable Spaces

I am having trouble with a practice exam question: $$\text{Show that if $X$ and $Y$ are metrizable, then so is $X\times Y$}$$ What I have so far: Given metric spaces $(X,d_x)$ and $(Y,d_y)$, I know ...
4
votes
3answers
67 views

The quotient map $q: \mathbb R^{n+1} \setminus \{0\} \to \mathbb P^n$ is open.

I'd like to show that the quotient map $q: \mathbb R^{n+1} \setminus \{0\} \to \mathbb P^n$ is open, where I'm considering $\mathbb P^n$ as the quotient space of $\mathbb R^{n+1} \setminus \{0\}$ ...
1
vote
0answers
31 views

Modes of convergence for continuous functions

I just wondered about what modes of convergence for continuous functions $f_n:X\rightarrow Y$ between topological spaces there are. Of course there is pointwise convergence, which is defineable for ...
0
votes
0answers
25 views

To prove existence of an open set of functions

I am trying to prove the following: In $C(X,Y)$ with $X=[0,1]$ and $Y$ of finite dimension $K$, $C(X,Y)$ having the topology of uniform convergence, for any $K$ finite there exists an open set of ...
1
vote
0answers
21 views

simplicial approximation and infinite complexes

It is well known that if $X$ is a finite simplicial complex then for every continuous map $f:|X|\to |Y|$ there exists a simplicial map $F: X^{(n)}\to Y$ that $|F|$ is homotopic to $f$. Does anyone ...
2
votes
2answers
70 views

General topology problem

Let $X$ be an infinite set. Let $T$ be a topology on $X$ such that all infinite subsets of $X$ belong to $T$. Prove that $T$ is the discrete topology on $X$. i know that all the infinite subsets of X ...
1
vote
1answer
24 views

James' theorem—going from the separable case to the general case

Consider the following famous theorem by Robert C. James (1964): Let $X$ be a Banach space over $\mathbb R$ and $C$ a non-empty, bounded, weakly closed subset. Then, $C$ is weakly compact if and ...
0
votes
1answer
21 views

First countable space and $\tau \subset \tau^*$

Let $\tau$ and $\tau^*$ be topologies on X with $\tau$ coarser than $\tau^*$ i.e $ \tau \subset \tau^*$. (i) Show that if (X,$\tau^*$) first countable, then (X,$\tau$) is also first countable I have ...
1
vote
0answers
45 views

Convention of a continued fraction presentation of a lens space

I want to clarify the following two conventions on a surgery description of a lens space. Let $p$ and $q$ are relatively prime integers. Express $$ ...
0
votes
1answer
19 views

$X$ contains at least two points & at least one isolated point. Prove $X$ is not connected.

Can we take two sets $G_1 = (x_1)$, where $x_1$ is the isolated point, and $G_2 = B(x_2;\epsilon)-(x_2)$ where $x_2$ is a limit point and show that the set- connectedness conditions hold? Help would ...
1
vote
1answer
23 views

Locally Compact Spaces: Separation Property

Given a locally compact Hausdorff space. Every compact set has a compact neighborhood base: $$C\subseteq U:\quad N\subseteq U\quad(C\subseteq N^°)$$ My construction was contrary to Rudin's: ...
1
vote
2answers
35 views

To determine whether range of f is closed , connected etc

Let $E= \{ (x,y) : |x| + |y| \leq 1 \}$ . Define $f : E \to \mathbb R$ by $f(x, y) = x + y / 1 + x^{2} + y^{2} $ Then range of $f$ is A . Connected open set B . Connected closed set C. ...
0
votes
4answers
75 views

Why does a topology contains its basis?

The definition of a basis is given as: Suppose a collection $\mathscr B$ of subsets of a given set $X$ satisfies: $\forall x \in X, \exists B \in \mathscr{B}$ such that $x \in B$. $B_1 , B_2 \in ...
3
votes
1answer
71 views

Homeomorphisms of an open set onto itself.

Given two distinct points $a,b\in U$ where $U$ is an open subset of $\mathbb R^n$, can one always find a homeomorphism $f:U\to U$ with $f(a)=b$ or does one need any further conditions on $ U $ or the ...
5
votes
3answers
71 views

How many of these are topologies?

Let $X$ be a set with $3$ elements. The set of subsets of the power set of $X$ is $2^{2^3}$ elements. How many of these are topologies? Is there a trick to this problem, or is it just a "plug ...
6
votes
1answer
66 views

Convergence of a sequence with assumption that exponential subsequences converge?

Problem One of my best friends asked me to think about the following problem: Suppose a sequence $\{a_n\}_{n=1}^\infty$ satisfies $\lim_{n\to\infty}a_{\lfloor\alpha^n\rfloor}=0$ for each ...
2
votes
1answer
99 views

Are all functions $f:\mathbf{Z}\to\mathbf{Z}$ “continuous”?

I read the following definitions in Glen E. Bredon's "Topology and Geometry": Let $\mathbf{x},\mathbf{y}\in\mathbf{R}^n$ and $$ ...
8
votes
4answers
282 views

An Illustrated Classification of Knots.

Let me be honest here: I know very little about Knot Theory. I'm sorry. I've a friend though, someone with no training in Mathematics at all but who is a huge fan of knots (for whatever reason), who ...
4
votes
1answer
84 views

Metrics on $\mathbb R^n$, Counting continuous functions and Open sets

Given the set $\mathbb{R}^n$ with metric $d$. We define continuous functions from $\mathbb{R}^n$ to $\mathbb{R}^n$ by open sets -we say that function is continuous iff the pre-image of every open set ...