Trait Suggestion: Long rays/lines S38, S39, S153 are Cozero complemented P61

[Moniker1998]

Trait Suggestion

Long rays/lines S38, S39, S153 are Cozero complemented P61, but this fact is not known to pi-Base today:
link to pi-Base 1
link to pi-Base 2
link to pi-Base 3

Proof/References

Even continuous function $f:X\to \mathbb{R}$ is eventually constant. If $U\subseteq X$ is a cozero-set, then either $U$ is bounded from above, or $U$ contains all values larger than $x$ for some $x\in X$.

In $U$ is bounded from above, say $U\subseteq \{y\in X : y < x\}$, then since $\{y\in X : y < x\}$ is homeomorphic to a subspace of $\mathbb{R}$, its cozero complemented, so there is open $V\subseteq \{y\in X : y < x\}$ disjoint from $U$ such that $U\cup V$ is dense in $\{y\in X : y < x\}$. If $V_0 = V\cup \{y\in X : y > x\}$ then $U, V_0$ are disjoint cozero sets and $U\cup V_0$ is dense in $X$.

If $U$ contains $\{y\in X : y > x\}$, let $U_0 = U\cap \{y\in X : y < x\}$ and find open $V\subseteq \{y\in X : y < x\}$ disjoint from $U_0$ such that $U_0\cup V$ is dense in $\{y\in X : y < x\}$. Then $U, V$ are disjoint cozero sets and $U\cup V$ is dense in $X$.

[Moniker1998]

As in #1044 those spaces are LOTS. This makes me think that perhaps every LOTS is cozero complemented.

[yhx-12243]

As in #1044 those spaces are LOTS. This makes me think that perhaps every LOTS is cozero complemented.

No. S141 (Ordered space $\omega_1 + 1 + \omega^*$) is a counterexample.

Clearly the $U = \omega^*$ is cozero. If $X$ is cozero complemented, then $V$ is a cozero subset of $\omega_1 + 1$, thus a cozero subset of $\omega_1$ (cozero set must be open).

However, cozero subset of $\omega_1$ are either bounded, are either containing the full tail. If bounded, then $U \cup V$ is not dense in $X$. If containg the full tail, then it containing the middle “1”, both into a contradiction.