MATHEMATICS

On tightness and pseudocharacter of compact $T_1$-spaces

A. A. Gryzlov, R. A. Golovastov

Udmurt State University, ul. Universitetskaya, 1, Izhevsk, 426034, Russia

Abstract: We consider the relationship between the pseudocharacter $\psi(X)$ and the tightness $t(X)$ of compact $T_1$-spaces $X$. We prove that $t(X)\leqslant\psi(X)$ for self-adjoined $T_1$-spaces, i.e., the spaces where a subset is closed if and only if it is compact. We also show that in general for compact $T_1$-spaces there is no relationship between these cardinal invariants. We give an example of a compact $T_1$-space such that the tightness of this space is uncountable, but its pseudocharacter is countable. Another example shows the space $X$ whose tightness is countable, but its pseudocharacter is uncountable.

Keywords: $T_1$-space, compact, tightness, pseudocharacter.

UDC: 515.122.22, 515.122.252

MSC: 54D10, 54D30

Received: 15.07.2019

DOI: 10.20537/vm190302

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