Definition:Altered Long Line

Definition

Let $\struct {L, \preccurlyeq, \tau}$ denote the long line.

Let $p$ be a point which is not an element of $L$.

Consider the set $S := L \cup \set p$.

Let $T = \struct {S, \tau_p}$ be the topological space whose topology $\tau_p$ is defined as follows:

The open sets of $T$ are the open sets of $L$ together with those generated by the following neighborhoods of $P$:

$\ds \map {U_s} p := \set p \cup \set {\bigcup_{\alpha \mathop = \beta}^\Omega \openint \alpha {\alpha + 1}: 1 \le \beta < \Omega}$

where $\Omega$ denotes the first uncountable ordinal.

$\map {U_s} p$ is therefore a right-hand ray without the ordinals.

The topological space $T = \struct {S, \tau_p}$ is known as the altered long line.

$p$ is the greatest element of $S = L \cup \set p$.