Topological Vector Spaces

Let $X$ be a set. A property of subsets of $X$ is a set $P\subset \mathcal{P}(X)$. We say that a subset $A\subset X$ has the property $P$, if $A\in P$. A property $P$ of subsets of $X$ is said to be stable under arbitrary intersections, if for every family $F$ of subsets of $X$ with property $P$, the intersection $\bigcap F$ has the property $P$. In other words, $P$ is stable under arbitrary intersections iff $\bigcap F\in P$ for every subset $F\subset P$. In this definition the family $F$ is allowed to be empty, hence $\bigcap \varnothing =X$ needs to have the property $P$.

For example, in a topological space $X$ the property of being a closed subset of $X$ is stable under arbitrary intersections.

If $P$ is stable under arbitrary intersections, and $A$ is a subset of $X$, which may or may not have the property $P$, then we define the $P$-hull of $A$ to be the intersection of all supersets $B\supset A$ having have the property $P$. By definition, the $P$-hull of $A$ has the property $P$. Moreover, it is the smallest superset of $A$ with property $P$ in the following sense: If $B$ is any superset of $A$ with property $P$, then $B$ contains the $P$-hull of $A$.

For example, the “closed”-hull of a subset $A$ of a topological space is the closure of $A$.

There are the dual notions of being stable under arbitrary unions and $P$-core with obvious definitions.

Definition (Convex, Balanced, Absolutely Convex)

Let $X$ be a vector space over the field $\mathbb{K}$. A subset $A\subset X$ is said to be

• convex if
• balanced if
• absolutely convex if

These properties of subsets of $X$ are stable under arbitrary intersections. Thus, we obtain the notions of convex hull $\mathrm{co}A$, balanced hull $\mathrm{bal}A$, and absolutely convex hull $\mathrm{aco}A$.