Articles of general topology

About the equivalence of definitions of a Baire Space

My definition of Baire Space is the modern one A topological space $X$ is a Baire Space iff the intersection of a countable family of open dense everywhere subsets of $X$ is dense everywhere. There is an old definition that states A topological space $X$ is a Baire Space iff every non empty open subset […]

How to show $X=\{A\in\mathcal{L}(\mathbb R^m, \mathbb R^n); \textrm{Ker}(A)=\{0\}\}$ is open in $\mathcal{L}(\mathbb R^m, \mathbb R^n)$?

How to show $X=\{A\in\mathcal{L}(\mathbb R^m, \mathbb R^n); \textrm{Ker}(A)=\{0\}\}$ is open in $\mathcal{L}(\mathbb R^m, \mathbb R^n)$? Here $\mathcal{L}(\mathbb R^m, \mathbb R^n)$ is the set of all linear applications between $\mathbb R^m$ and $\mathbb R^n$.

Positive definite part of a symmetric matrix – or: are the positive definite matrices a retract of the set of symmetric matrices?

$\newcommand{\Sym}{\operatorname{Sym}}$ Denote by $\Sym(n)$ the set of symmetric, real $n\times n$ matrices and let $\iota:\Sym^+(n)\hookrightarrow \Sym(n)$ be the subset of positive definite matrices with its standard topologies. My question: is there a continuous map $r:\Sym(n)\to \Sym^+(n)$ with $r\circ\iota=\operatorname{id}_{\Sym^+(n)}$ (a retraction), i.e. can we speak of a positive definite part $r(A)$ of a symmetric matrix $A$?

Compactifications of limit ordinals

I thought I knew that but it seems I don’t. Let $\alpha$ be a countable, limit ordinal $\alpha>\omega$. Give $\alpha$ its order topology. What is the Stone-Čech compactification of $\alpha$? Is there any reason why it should be $\beta \omega$?

Fundamental group of the quotient of a cylinder by a rotation at either end

The following is a past qual problem: let $X = S^1 \times [0,1]$ be the cylinder, and define an equivalence relation on $X$ by $(z,1) \sim (iz,1)$ and $(w,0) \sim (e^{i\pi /7} w, 0)$. Compute $\pi_1(X/{\sim})$. I’ve been trying to realize a CW complex structure for $X/{\sim}$, by having a 2-cell with edges $a^4$ at […]

compactness in topology of pointwise convergence

I started reading about the topology of pointwise convergence. So far I do not feel quite comfortable with this theory. Maybe one can help me out in a more concrete example case. Let’s consider sequences, so $a : \mathbb{N} \to \mathbb{R},x \mapsto a(x)$. In the topology of pointwise convergence it holds now that $$(a)_n \to […]

Uncountable disjoint union of measure spaces

Let $(a,b)$ be an interval. Let $(A_i, \Sigma_i, \mu_i)$ be a measure space for each $i \in (a,b)$. Is it possible to put a measure space on the disjoint union $$\bigcup_{i \in (a,b)}\{i\}\times A_i?$$ If the $A_i = \Omega_i$ were open subsets of $\mathbb{R}^n$, we can think of this disjoint union as a non cylindrical […]

Convex functions and uniform convergence of derivatives

Let $f_n:[0,1]\to\mathbb{R},\ n\in\mathbb{N}$ be a sequence of convex analytic functions. Consider the sequence of derivatives $(f_n’)_{n\in\mathbb{N}}$. Suppose that $$f’_n(x)\xrightarrow[n\to\infty]{}g(x)\in\mathbb{R}\ \text{for all}\ x\in\mathbb{Q\cap[0,1]}.$$ Is it true that $(f_n’)_{n\in\mathbb{N}}$ converges uniformly on $\mathbb{Q}\cap[0,1]$?

Precise definition of epsilon-ball

My textbook gives the following definition: “For each $\epsilon>0$, the $\epsilon$-ball about a point $x$ in a metric space $M$ is the set $\{y\in M:d(x,y)<\epsilon\}$.” Is this correct? Because this sounds as if $\epsilon$ is just a dummy variable, and that there is such a thing as, say, a “5-ball” meaning $\{y\in M:d(x,y)<5\}$. Shouldn’t the […]

About measure theoretic interior and boundary

Reference:- Evans-Gariepy, Federer, other books and notes of geometric measure thoery. I just want to clarify whether these definitions of measure theoretic interior and boundary are correct. Given a Borel regular measure $ \mu $ in $\mathbb{R}^n $, given a $\mu$-measurable subset $E \subset \mathbb{R}^n $, let $$ \psi(x,E) = \lim_{r\rightarrow 0}\frac{\mu(E\cap B(x,r))}{\mu(B(x,r))} $$ Here […]