I'm trying to show that the Least Core of $\displaystyle \Pi$ is a convex set whenever $\displaystyle \Pi$ is a convex and closed set in a topological vector space.

The definition of the Least Core is:

$\displaystyle LC(\Pi,F):=\{x \in\Pi | \bigvee_{j\in M}F_j(x)\leq\bigvee_{j\in M}F_j (y) \forall y\in \Pi\}$ Where $\displaystyle \bigvee$ is the maximum operator and M is the index set of all the funtions on $\displaystyle \Pi$

But what is the definition of a convex subset of a topological space that I should use?

Can I just use this notion:

Let $\displaystyle V$ be a vector space (over R or C). A subset S of V

is convex if for all points x,y in S, the line segment

$\displaystyle \{\alpha x + (1-\alpha) y \mid \alpha\in(0,1)\}$ is also in S.

Btw this is no homework, so I don't know exactly what definition to use.