Articles of the baire space

$\Sigma_{\alpha+1}^0(X)$ universal set based on the Baire space

In my attempts to write a cleaner proof (compared to what I was taught) to the fact that for uncountable Polish spaces the Borel hierarchy does not stop until $\omega_1$ I am trying to prove the following specific case: Suppose $X$ is a perfect Polish space, and $G_\alpha\subseteq\omega^\omega\times X$ is a universal set for $\Pi_\alpha^0(X)$ […]

Dense subset of Cantor set homeomorphic to the Baire space

Does anyone know a proof that the Cantor set, $\{0,1\}^{\mathbb{N}}$, has a dense subset homeomorphic to the Baire space, $\mathbb{N}^{\mathbb{N}}$? Thank you.

Proof that $\omega^\omega$ is completely metrizable and second countable

I have almost solved the following problem but am stuck at the very end, can you help me finish it? Thank you for your help. Let $n<\omega$ and $t\in {}^n\omega$. We define $U_t=\{s\in {}^\omega\omega : t\subseteq s\}$. The family $\mathcal B=\{U_t : t\in\bigcup {}^n\omega\}$ is a basis for a topology $\tau$ on ${}^\omega\omega$. This means […]

Basic questions about $\mathbb{Z}^{\mathbb{N}}$ with the product topology

can someone please let me know if the following is correct: 1) Let $\mathbb{Z}$ be the integers endowed with the discrete topology and $\mathbb{N}$ the natural numbers. Is $\mathbb{Z}^{\mathbb{N}}$ a discrete space with the product topoogy? 2) Does $\mathbb{Z}^{\mathbb{N}}$ contain a compact infinite set? 3) Is $\mathbb{Z}^{\mathbb{N}}$ metrizable? My work: 1) I think this is […]

Is the Baire space $\sigma$-compact?

Is the Baire space $\sigma$-compact? The Baire space is the set $\mathbb{N}^\mathbb{N}$ of all sequences of natural numbers under the product topology taking $\mathbb{N}$ to be discrete. It is a complete metric space, for example with the metric $d ( x , y ) = \frac{1}{n+1}$ where $n$ is least such that $x(n) \neq y(n)$. […]

Baire space homeomorphic to irrationals

I try to show that the Baire space $\Bbb N^{\Bbb N}$, with regular product metric, is homeomorphic to the unit interval of irrationals $(0,1)\setminus\Bbb Q$. I already know that the needed function is using continued fractions $$a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3+\cfrac1{a_4+\cfrac1\ddots }}}}$$ My question is how to show that this function actually fulfills […]

Prove that Baire space $\omega^\omega$ is completely metrizable?

When I tried to prove that Baire space $\omega^\omega$ is completely metrizable, I defined a metric $d$ on $\omega^\omega$ as: If $g,h \in \omega^\omega$ then let $d(g,h)=1/(n+1)$ where $n$ is the smallest element in $\omega$ so that $g(n) \ne h(n)$ is such $n$ exists, and $d(g,h)=0$ otherwise. I am stuck trying to prove this metric […]