The ring of real-valued functions which are continuous on a dense cozero set
Amrita Dey, Sagarmoy Bag, Dhananjoy Mandal
TL;DR
The paper introduces and studies the ring $T''(X)$ of real-valued functions on a space $X$ that are continuous on a dense cozero set, situating it between $C(X)$ and $T'(X)$. It provides precise characterizations for when $T''(X)=C(X)$ (precisely for Tychonoff $X$ that are almost $P$-spaces) and investigates when $T''(X)$ is isomorphic to $C(Y)$, showing obstructions in general. It develops a rich algebraic framework for $T''(X)$, including a full description of $z$-ideals, the socle, Noetherian/Artinian criteria tied to the finiteness of $X$, and a comprehensive set of equivalent conditions for $T''(X)$ to be Von Neumann regular. The paper also introduces nowhere almost $P$-spaces, giving topological characterizations and linking them to the inclusion $C(X)_F\subseteq T''(X)$ and to the property that $C(X)=T''(X)$ only when $X$ is discrete, and ends with open questions guiding future work.
Abstract
Let $T''(X)$ and $T'(X)$ denote the collections of all real-valued functions on $X$ which are continuous on a dense cozero set and on an open dense subset of $X$ respectively. $T''(X)$ contains $C(X)$ and forms a subring of $T'(X)$ under pointwise addition and multiplication. We inquire when $T''(X)=C(X)$ and when $T''(X)=T'(X)$. We also ponder over the question when is $T''(X)$ isomorphic to $C(Y)$ for some topological space $Y$. We investigate some algebraic properties of the ring, $T''(X)$ for a Tychonoff space $X$. We provide several characterisations of $T''(X)$ as a Von-Neumann regular ring. We define nowhere almost $P$-spaces using the ring $T''(X)$ and characterise it as a Tychonoff space which has no non-isolated almost $P$-points. We show that a Tychonoff space with countable pseudocharacter is a nowhere almost $P$-space and highlight that this condition is not superflous using the closed ordinal space.