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The long ray (S38) is the lexicographic order topology on $Z=\omega_1\times[0,1)$. The open long ray (S153) is the subspace $Z\setminus\{\langle 0,0\rangle\}$ of $Y$. The closed long ray (S39) is the subspace $Y=(\omega_1\times[0,1))\cup\{\langle\omega_1,0\rangle\}$ of the lexicographic order topology on $(\omega_1+1)\times[0,1)$. Finally, the long circle (S196) is the quotient $X=Y/\{\langle0,0\rangle,\langle\omega_1,0\rangle\}$ identifying the least and greatest points of the closed long ray.

As linearly ordered topological spaces (LOTS) (P133), it's immediate that the [closed/open] long ray is $T_5$, that is, completely/hereditarily normal and $T_1$. Can we say the same for the long circle?

Steven Clontz
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    Thank you for including the definitions of all your terms! I can't answer your questions, but I enjoy seeing all the wild topological spaces. – JonathanZ Dec 31 '24 at 16:25
  • Maybe I haven't followed your definitions closely, but $X$ seems to be the closed image of $Y$, hence monotonically normal (hence hereditarily collectionwise normal and hereditarily countably paracompact). – Tyrone Jan 02 '25 at 14:52

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We can. One way to see this is to observe that by removing a single point of the long circle, we obtain a copy of either the open long ray (by removing the identified point), or the linearly ordered topological space $\big((\omega_1+1)\times[0,1)\big)\setminus\{\langle 0,0\rangle\}$ (by removing any other). It follows that every proper subspace of the long circle is a subspace of a LOTS. Since LOTS are hereditarily normal, every proper subspace of the long circle is normal.

Finally, we need to show that the entire long circle is normal. This may be observed by the fact that $T_4$ (normal Hausdorff) is preserved for quotients identifying a closed set $C$ to a point, see e.g. If $X$ is normal and $A \subset X$ is closed, then the quotient space $X/A$ is normal..

Steven Clontz
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