Compact images of spaces with a weaker metric topology

Yan, Peng-fei ; Lü, Cheng

Czechoslovak Mathematical Journal, Tome 58 (2008) no. 4, pp. 921-926.

Résumé

If $X$ is a space that can be mapped onto a metric space by a one-to-one mapping, then $X$ is said to have a weaker metric topology. \endgraf In this paper, we give characterizations of sequence-covering compact images and sequentially-quotient compact images of spaces with a weaker metric topology. The main results are that \endgraf (1) $Y$ is a sequence-covering compact image of a space with a weaker metric topology if and only if $Y$ has a sequence $\{\mathcal F_i\}_{i\in \mathbb N}$ of point-finite $cs$-covers such that $ {\bigcap _{i\in \mathbb N}}\mathop{\rm st} (y,\mathcal F_i)=\{y\}$ for each $y\in Y$. \endgraf (2) $Y$ is a sequentially-quotient compact image of a space with a weaker metric topology if and only if $Y$ has a sequence $\{\mathcal F_i\}_{i\in \mathbb N}$ of point-finite $cs^*$-covers such that ${\bigcap _{i\in \mathbb N}}\mathop{\rm st} (y,\mathcal F_i)=\{y\}$ for each $y\in Y$.

Détail

Classification : 54C10, 54C40, 54E99
Keywords: sequence-covering mappings; sequentially-quotient mappings; compact mappings; weaker metric topology