TITLE: Question about pointwise canonically weakly pseudocompact space.
QUESTION [6 upvotes]: 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.

REPLY [5 votes]: @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.