A theorem on the cardinality of $\kappa$-total spaces

Abstract:

Throughout this article, $\kappa$ denotes an arbitrary infinite cardinal number. In 1979, A. A. Gryzlov strengthened a well-known result of A. V. Arhangel’skii by proving that every compact ${T_1}$-space of pseudocharacter $\kappa$ has cardinality $\leqslant {2^\kappa }$. Using techniques similar to Gryzlov’s, we prove that every ${2^\kappa }$-total, ${T_1}$-space of pseudocharacter $\leqslant \kappa$ is compact and hence of cardinality $\leqslant {2^\kappa }$. Some related results and examples are given.