A point $x$ of a space $X$ is said to be a point of canonical weak pseudocompactness if the following condition is satisfied:
For every canonical open subset $U$ of $X$ such that $x\in\overline{U}$, there exists a sequence $\{A_n:n\in\omega\}$ of subsets of $U$ such that $x\in\overline{A_n}$, for each $n\in \omega$, and for every indexed family $\xi=\{O_n:n\in\omega\}$ of open subsets of $X$ satisfying $O_n\cap A_n\neq\varnothing$ for all $n\in\omega$, the family $\xi$ has an accumulation point in $X$.
A space $X$ is pointwise canonically weakly pseudocompact if each point of $X$ is a point of canonical weak pseudocompactness.
How can we show that Every compact topological group is a pointwise canonically weakly pseudocompact space?
Is it true that every dyadic compactum is a pointwise canonically weakly pseudocompact space?
If 2 is true then 1 has also proven since Every compact topological group $G$ is a dyadic compactum.
An open subset $U$ of a space $X$ is said to be canonical open in $X$ if $U$ is the interior of its closure.
I added a few references that I hope will clarify the issue;
1. A.V. Arkhangel'skii and M. Tkachenko, Topological Groups and Related Structures, p359.
2. A.V. Arhangel’skii, Moscow Spaces and Topological Groups, p404.
3.A.V. Arhangel'skii, On a theorem of W.W. Comfort and K.A. Ross, p141.
@TXC: This questions seems to be trivial (or I misunderstood it). If a space $X$ is pseudocompact (a feebly compact, in Tkachenko’s terminology), then, by the definition of a pseudocompact space, the family $\xi$ cannot be locally finite. In particular, each compact space is pointwise canonically weakly pseudocompact.