If $X$ is a convex (closed) set and if $\text{int}\,X_i=\text{int}\,X$ and $\text{cl}\,X_i=\text{cl}\,X$ then is $\bigcup_{i\in I} X_i$ convex?

Question

Let be $V$ a vector topological space and so let be $X$ a convex (closed) set so that we suppose that $\mathfrak X:=\{X_i\subseteq X:\,i\in I\}$ a collection of convex subsets of $X$ such that $$ \operatorname{int}X_i=\operatorname{int} X\,\,\,\text{and}\,\,\,\operatorname{cl}X_i=\operatorname{cl}X $$ for each $i\in I$. So with this hypotheses is the set $\bigcup\mathfrak X$ convex? In particular if the statement is generally false is true when $X$ is closed? Could someone help me, please?

Answer 1

Consider a triangle in the plane. $X$ is the triangle together with its interior. $X$ is a closed convex set. Let $X_1$ be the interior of the triangle together with one vertex. Let $X_2$ be the interior of the triangle together with another vertex.