On Functorial Lindelöfifiability

Abstract

In the present paper, we prove that a topological space admits a functorial Lindelöfification if and only if its realcompactification is Lindelöf. To investigate the functorial Lindelöfifiability of a topological space, for each topological property $\mathsf{P}$, we introduce the notion of "functorial $\mathsf{P}$-ification" and give an explicit construction of the functorial $\mathsf{P}$-ification. Moreover, for a discrete space $X$, we discuss the functorial $|X|$-Lindelöfifiability of $X$ and study relationships with properties of the cardinal $|X|$. Finally, we apply our results concerning functorial $\kappa$-Lindelöfifiability (for some cardinal $\kappa$) to the space of ordinals and construct several functorial $\kappa$-Lindelöfifiable spaces.

11 pages