Increasing chains and discrete reflection of cardinality

TL;DR

This paper refines a classical theorem by establishing a cardinal inequality for increasing unions of specific topological spaces and addresses a problem on the reflection of cardinality by closures of discrete sets.

Contribution

It introduces a new cardinal inequality for unions of strongly discretely Lindelof spaces and provides partial results on the reflection of cardinality by discrete set closures.

Findings

Cardinality of certain unions is at most continuum.

Partial positive answer to Dow's problem on reflection of cardinality.

Refinement of Arhangel'skii Theorem for specific space classes.

Abstract

Combining ideas from two of our previous papers, we refine Arhangel'skii Theorem by proving a cardinal inequality of which this is a special case: any increasing union of strongly discretely Lindelof spaces with countable free sequences and countable pseudocharacter has cardinality at most continuum. We then give a partial positive answer to a problem of Alan Dow on reflection of cardinality by closures of discrete sets.