$|F\cap K^{-1}A|\leq \sum_{k\in K}|F\cap k^{-1}A|$?, where $K$ is compact set, $F$ is finite set in topological semigroup $T$

Question

Let $T$ be a topological semigroup and $F\subseteq T$ be a non-empty finite set, $K\subseteq T$ be a compact set and $A$ be a non-empty subset in $T$. What can say about relation between $|F\cap K^{-1}A|$ and $\sum_{k\in K}|F\cap k^{-1}A|$? Is it true that $|F\cap K^{-1}A|\leq \sum_{k\in K} |F\cap k^{-1}A|$?

Where

$K^{-1}A= \{t\in T: Kt\cap A\neq \emptyset\}$

Answer 1

Several assumptions are superfluous. Indeed the inequality $$ |F\cap K^{-1}A| \leqslant \sum_{k\in K} |F\cap k^{-1}A| \quad (*) $$ holds for any subsets $A$, $F$ and $K$ of a semigroup $T$ (no topological assumption required). I let you verify that $$ K^{-1}A = \bigcup_{k \in K} k^{-1}A $$ It follows by distributivity that $$ F \cap K^{-1}A = \bigcup_{k \in K}\ (F \cap k^{-1}A) $$ which gives immediately $(*)$.