Algebraic closure of a subset formula_1 of a vector space formula_2 is the set of all points that are linearly accessible from formula_1. It is denoted by formula_4 or formula_5.
A point formula_6 is said to be linearly accessible from a subset formula_7 if there exists some formula_8 such that the line segment formula_9 is contained in formula_1.
Necessarily, formula_11 (the last inclusion holds when "X" is equipped by any vector topology, Hausdorff or not).
The set "A" is algebraically closed if formula_12.
The set formula_13 is the algebraic boundary of "A" in "X".
Examples.
The set formula_14 of rational numbers is algebraically closed but formula_15 is not algebraically open
If formula_16 then 
formula_17. In particular, the algebraic closure need not be algebraically closed.
Here, formula_18.
However, formula_19 for every finite-dimensional convex set "A".
Moreover, a convex set is algebraically closed if and only if its complement is algebraically open.