An Extension of Baire's Category Theorem to Relator Spaces

Árpád Száz

As a particular case of a more general result, we show that if $\mathcal{R}$ is a topological, topologically filtered, topologically regular relator on $X$ such that $\mathcal{R}$ is either topologically relatively locally sequentially compact, or uniformly countable and properly sequentially convergence-adherence complete, then $\mathcal{R}$ is a Baire relator. If $X$ is a nonvoid set, then by a relator $\mathcal{R}$ on $X$ we mean a nonvoid family of binary relations on $X$. The relator $\mathcal{R}$ is called a Baire relator if the fat subsets of the relator space $X(\mathcal{R})$ are not meager. A subset $A$ of $X(\mathcal{R})$ is called fat if $\mathrm{int}_{\mathcal{R}}(A)\neq\emptyset.$ While, the set $A$ is called meager if it is a countable union of rare (nowhere dense) sets.