Definition:Tychonoff Corkscrew

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Let $\omega$ be the first transfinite ordinal.

Let $\Omega$ be the first uncountable ordinal.

For each ordinal $\alpha$, let $A_\alpha$ denote the totally ordered set defined as:

$A_\alpha := \set {-0, -1, -2, \ldots, \alpha, \ldots, 2, 1, 0}$

with the order topology.

One instance of the plane $P^*$

Let $P$ be the product space defined as:

$P := A_\Omega \times A_\omega$

Let $P^*$ be the topological subspace of $P$:

$P^* := P \setminus \set {\tuple {\Omega, \omega} }$

Consider an infinite stack of copies of $P^*$.

Let a rectangular corkscrew lattice $C$ be formed such that it spirals in both directions by:

$(1): \quad$ cutting each $P^*$ immediately below the positive $A_\Omega$ axis $\tuple {\Omega, \omega}$ to $\tuple {0, \omega}$
$(2): \quad$ joining the fourth quadrant of one $P^*$ to the first quadrant of the $P^*$ immediately below it, along the positive $A_\Omega$ axis.

Let $\sequence {\map { {A_\Omega}^+} i}_{i \mathop = -\infty}^{+ \infty}$ be the family of positive $A_\Omega$ axes.

Then $i$ will be referred to as the level of $A_\Omega$.

By convention, the points of $C$ which lie above $\map { {A_\Omega}^+} i$ are at a level greater than $i$.

If $x$ is such a point, this is denoted $\map L x > i$.

We create $S$ by adding to $C$ two distinguished points $a^+$ and $a^-$ which can be considered as being infinity points at the top and bottom of the axis of the corkscrew.

Basis neighborhoods of $a^+$ consist of all points of $S$ which lie above a certain level.

Basis neighborhoods of $a^-$ consist of all points of $S$ which lie below a certain level.

The topological space $\struct {S, \tau}$ so generated is referred to as the Tychonoff corkscrew.

Deleted Tychonoff Corkscrew

Let $T = \struct {S, \tau}$ denote the Tychonoff corkscrew.

Let $a^-$ denote the distinguished point of $S$ considered as being the infinity point at the bottom of $T$.

The deleted Tychonoff corkscrew is the topological subspace defined as:

$T_\infty = \struct {S \setminus \set {a^-}, \tau}$

Also see

  • Results about the Tychonoff corkscrew can be found here.

Source of Name

This entry was named for Andrey Nikolayevich Tychonoff.