$\theta$-regularity in spaces with more topologies

conference paper

English

The term {\it space} $(X,\tau,\sigma,\rho)$ is referred as a set $X$ with three, generally non-identical topologies $\tau$, $\sigma$ and $\rho$. We say that $x\in X$ is a {\it $(\sigma, \rho)$-$\theta$-cluster point} of a filter base $\Phi$ in $X$ if for every $V\in\sigma$ such that $x\in V$ and every $F\in\Phi$ the intersection $F\cap\cl_\rho V$ is non-empty. If $\Phi$ has a cluster point with respect to the topology $\tau$, we say that has a {\it $\tau$-cluster point}. \medskip \parindent=0pt {\bf Definition 1.} We say that a space $X$ is said to be {\it(countably) $(\tau,\sigma,\rho)$-$\theta$-regular} if every (countable) $\tau$-closed filter base $\Phi$ with a $(\sigma,\rho)$-$\theta$-cluster point has a $\tau$-cluster point. \medskip {\bf Theorem A.} {\sl Let $X$ be the product (sum) space for the family $\left\{X_\iota |\iota\in I\right\}$ of $(\tau_\iota,\sigma_\iota,\rho_\iota)$-$\theta$-regular spaces $X_\iota$ with the corresponding product (sum) topologies $\tau$, $\sigma$, $\rho$. Then $X$ is $(\tau,\sigma,\rho)$-$\theta$-regular.} \medskip {\bf Defintion 2.} A bitopological space $(X,\tau,\sigma)$ is said to be {\it $\beta$-pairwise (countably) $\theta$-regular} if $X$ is (countably) $(\tau,\sigma,\tau)$-$\theta$-regular and (countably) $(\sigma,\tau,\sigma)$-$\theta$-regular. We say that the bitopological space $X$ is \it {$\delta$-pairwise (countably) $\theta$-regular} if $X$ is (countably) $(\tau\vee\sigma,\sigma,\tau\vee\sigma)$-$\theta$-regular and (countably) $(\tau\vee\sigma,\tau,\tau\vee\sigma)$-$\theta$-regular. \medskip {\bf Theorem B.} {\sl Let $X$ be a bitopological space. Then $X$ is RR-pairwise (FHP-pairwise) paracompact \iff $X$ is $\beta$-pairwise countably $\theta$-regular and RR-pairwise (FHP-pairwise) semiparacompact.} \medskip {\bf Theorem C.} {\sl Let $X$ be $\delta$-pairwise countably $\theta$-regular. Then $X$ is $\delta$-pairwise paracompact \iff $X$ is $\delta$-pairwise semiparacompact.}

More topologies on a set, $\theta$-regularity, pairwise paracompactness.

KOVÁR, M.

7. 4. 1999

Escuela Universitaria de Gandia

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