Definition:Hewitt's Condensed Corkscrew

From ProofWiki
Jump to navigation Jump to search

Definition

Let $\omega$ be the first transfinite ordinal.

Let $\Omega$ be the first uncountable ordinal.

Let $\hointr 0 \Omega$ denote the set of countable ordinals.

Let $T = \struct {S, \tau}$ be the Tychonoff corkscrew:

$S = C \cup \set {a^+} \cup \set {a^-}$

where $C$, $a^+$ and $a^-$ are as in that definition.

Let $A = T \times \hointr 0 \Omega$.

Let $S$ be the subset of $A$ consisting of $C \times \hointr 0 \Omega$.


Hence we can consider $A$ as being an uncountable sequence of Tychonoff corkscrews $A_\lambda$ where $\lambda \in \hointr 0 \Omega$.

Similarly, we can consider $S$ as being that same uncountable sequence of Tychonoff corkscrews missing all those distinguished points $a^+$ and $a^-$ at infinity.


Let $\Gamma: S \times S \to \hointr 0 \Omega$ be a bijection.

Let $\pr_i$ for $i \in \set {1, 2}$ be the projections from $S \times S$ to $S$.

Let us define the mapping $\psi: A \setminus S \to S$ by:

\(\displaystyle \map \psi { {a^+}_\lambda}\) \(=\) \(\displaystyle \pr_1 \map {\Gamma^{-1} } \lambda\)
\(\displaystyle \map \psi { {a^-}_\lambda}\) \(=\) \(\displaystyle \pr_2 \map {\Gamma^{-1} } \lambda\)

Then for $2$ distinct points $x, y \in S$, there exists some $\lambda \in \hointr 0 \Omega$, that is $\lambda = \Gamma {x, y}$, such that both $\psi^{-1} \sqbrk x$ and $\psi^{-1} \sqbrk y$ intersect $A_\lambda$.


The topology $\tau$ on $A$ is determined by the basis neighborhoods $N$ of each $x \in S$ with the property that:

$\psi^{-1} \sqbrk {N \cap S} \subseteq N$

together with $A_\lambda$-basis neighborhoods called tails of each $a \in A \subseteq S$.

Hence $S$ will inherit the subspace topology from $A$.


A typical basis neighborhood of $x \in S$ is constructed as follows:

We begin by taking a $\sigma$-neighborhood $N_0$ of $x \cup \phi^{-1} \sqbrk x$ where $\sigma$ is the product topology on $A = T \times \hointr 0, \Omega$, where $\hointr 0, \Omega$ is given the discrete topology.

Then we define $N_i$ recursively as follows:

$N_i$ is a $\sigma$-neighborhood of $N_{i - 1} \cup \psi^{-1} \sqbrk {N_{i - 1} \cap S}$

and:

$N = \bigcup N_i$

Then $\psi^{-1} \sqbrk {N \cap S} \subseteq N$.


The topological space $\struct {S, \tau}$ so generated is referred to as Hewitt's condensed corkscrew.


Also see

  • Results about Hewitt's condensed corkscrew can be found here.


Source of Name

This entry was named for Edwin Hewitt.


Sources