$\theta$-regularity in spaces with more topologies

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}.

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{\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.

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{\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.}

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{\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.


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{\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.}

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{\bf Theorem C.} {\sl Let $X$ be $\delta$-pairwise countably $\theta$-regular.
Then $X$ is $\delta$-pairwise paracompact \iff $X$ is $\delta$-pairwise
semiparacompact.}