2025/02/21 by Dey, Amrita, Bag, Sagarmoy, Mandal, Dhananjoy
#FOS: Mathematics #General Topology (math.GN)
paper · doi:10.48550/arxiv.2502.15358
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.