Articles of general topology

A perfect Hausdorff space that is not metrizable.

Can anyone provide an example of a well-known topological space that has the following three properties: (1) It is perfect (contains no isolated points), (2) T2, and (3) not metrizable.

Is the topological product $\Pi \{G_\alpha: \alpha < \kappa\} $ a topological group?

Let $\mathcal G= \{G_\alpha: \alpha < \kappa\}$ be a family of topological groups. My question is this: Is the topological product $\Pi \{G_\alpha: \alpha < \kappa\} $ a topological group? Thanks ahead.

Closed maps in terms of “thickening fibers”

Recently, I have asked for intuition for the characterization of compactness of a space $X$ in terms of the closedness of the projection $X\times Y\rightarrow Y$. The wonderful answer I have received includes the following statment: A continuous function $f:X\rightarrow Y$ is closed iff for each $y\in Y$ and each open $U\subset X$ with $f^{−1}[{y}]\subset […]

Induced map on simplices from order-preserving maps between finite ordinal numbers

Let $\Delta$ be the category of finite ordinal numbers with order-preserving maps, i.e., $\Delta$ consists of objects strings $[n]: 0→1→2→⋯→n$. A morphism $f:[n]→[m]$ is an order-preserving function (a functor) and we can think of the morphism like diagrams where arrows don’t cross. For an arrow $[n] \overset{\Theta}{\to} [m]$ we get an induced map $| \Delta^n […]

Compactness and Limit points

If $A$ is compact how can it be shown that any sequence $\{x_n\}$ in $A$ has a limit point in $A$? I know this is proven in a lot of textbooks but I’m finding this hard to conceptualise.

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 […]