The Lω-Compactness in Lω-Spaces

Abstract

The concepts of $\alpha \omega$-remote neighborhood family, $\gamma \omega$-cover, and $\mathrm{L\omega }$-compactness are defined in $\mathrm{L\omega }$-spaces. The characterizations of $\mathrm{L\omega }$-compactness are systematically discussed. Some important properties of $\mathrm{L\omega }$-compactness such as $\omega$-closed heredity, arbitrarily multiplicative property, and preserving invariance under $\omega$-continuous mappings are obtained. Finally, the Alexander $\omega$-subbase lemma and the Tychonoff product theorem with respect to $\mathrm{L\omega }$-compactness are given.