Influence of ideals in compactifications

Abstract

One point compactification is studied in the light of ideal of subsets of ℕ. 𝓘-proper map is introduced and showed that a continuous map can be extended continuously to the one point 𝓘-compactification if and only if the map is 𝓘-proper. Nowhere tallness, introduced by P. Matet and J. Pawlikowski in [J. Symb. Log. 63(3) (1998), 1040–1054], plays an important role in this article to study various properties of 𝓘-proper maps. It is seen that one point 𝓘-compactification of a topological space may fail to be Hausdorff even if the underlying topological space is Hausdorff but a class {𝓘} of ideals has been identified for which one point 𝓘-compactification coincides with the one point compactification if the underlying topological space is metrizable. Let’s speak our minds that the results in this article will look elegant if one looks at it from a topological angle.