Question1. I can prove that Every metric space is Hausdorff. Where do they use this fact in this proof? Question2. Let $C$ be a closed subset of $X$ and $p\in X\setminus C$. How do I prove the function $f:X\to I=[0,1]$ by $f(x)=\min\{\frac{d(x,C)}{d(p,C)},1\}$. How do I prove that $f$ is conti...

I would like to know some non-trivial examples of partial combinatory algebras whose realizability universe does not satisfy all of the axioms of Constructive Zermelo-Fraenkel (CZF) set theory. Based on what I read in the notes Realizability by Streicher it seems that the realizability model of ...

Let $f \in L^1(\Bbb{T})$ and $Q_r(\theta)=\frac{2r\sin{\theta}}{1-2r \cos{\theta}+r^2},r \in[0,1]$ the conjugate Poisson kernels and $P_r(\theta)=\frac{1-r^2}{1-2r \cos{\theta}+r^2},r \in[0,1]$ and $\theta \in [0,2\pi]$ the Poisson Kernels. Fix $z_0=re^{i\theta} \in D=\{z:|z|<1\}$ We...