In the book Munkres: Topology §30 I met the following problem: Show that if $X$ has a countable basis $\{B_n\}_{n \in \mathbb{Z}_+}$, then every basis $\mathscr{C}$ for $X$ contains a countable basis for $X$. [Hint: For every pair of indeces $n$, $m$ for which it is possible, choose $C_{n, m}...

I've googled this and I've seen different types of proofs but they all use notations that I don't understand. But first of all, I need to understand what quadratic convergence means, I read that it has to do with the speed of an algorithm. Is this correct? Ok, so I know that this is the Newto...

Let $f(z)$ be analytic in a simple closed domain $D$ and on its boundary, the simple closed contour $C$. If $f(z)$ is injective on $C$, then $f(z)$ is injective on $D$. If $z_0\in D$ is a point such that $f(z_0)\notin f(C)$ then we can use an argument principle to conclude $f(z_0) = f(z)$ then ...

Prove that $\mathcal P((-1,1))$ and the set of all functions $f:\Bbb R\to\Bbb R$ which attain every value in $\Bbb Z$ uncountably many times are equinumerous. My attempt: My first claim: Let $S=\{A\subseteq\Bbb R\mid A\text{ is uncountable}\}.\operatorname{card}(S)=2^{\mathfrak c}.$ $\boxed{\l...