Definition:Hewitt's Condensed Corkscrew
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:
| \(\ds \map \psi { {a^+}_\lambda}\) | \(=\) | \(\ds \pr_1 \map {\Gamma^{-1} } \lambda\) | ||||||||||||
| \(\ds \map \psi { {a^-}_\lambda}\) | \(=\) | \(\ds \pr_2 \map {\Gamma^{-1} } \lambda\) |
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.
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Also see
- Results about Hewitt's condensed corkscrew can be found here.
Source of Name
This entry was named for Edwin Hewitt.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.): Part $\text {II}$: Counterexamples: $92$. Hewitt's Condensed Corkscrew