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.

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.

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 […]

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 […]

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.

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 $\mathcal{L}(\mathbb R^m, \mathbb R^n)$? Here $\mathcal{L}(\mathbb R^m, \mathbb R^n)$ is the set of all linear applications between $\mathbb R^m$ and $\mathbb R^n$.

$\newcommand{\Sym}{\operatorname{Sym}}$ 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 $A$?

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 $\beta \omega$?

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 […]

Intereting Posts

Finding determinant for a matrix with one value on the diagonal and another everywhere else
Some asymptotics for zeta function.
application of positive linear functionl
Integration (Fourier transform)
Sorting numbers in a matrix by moving an empty entry through other entries is not always possible .
Calculate position of rectangle lower left corner on center rotatiton
Finding matching roots
Simplifying $\sum_{r = 0}^{n} {{n}\choose{r}}r^k(-1)^r$
prove that a function is an immersion
Generalizing $\sum \limits_{n=1}^{\infty }n^{2}/x^{n}$ to $\sum \limits_{n=1}^{\infty }n^{p}/x^{n}$
Which coefficients of the characteristic polynomial of the shape operator are isometric invariants?
Is there always a telescopic series associated with a rational number?
Why it is absolutely mistaken to cancel out differentials?
Nilpotent matrix and basis for $F^n$
If $x_1 = 3$, $x_{n+1} = \frac{1}{4-x_n}$ for $n \geq 1$, prove the sequence is bounded below by $0$, above by $4$.