I am trying to prove that, the relation $R$, defined on a topological space $X$ s.t. "$aRb$ holds" iff "there exists a path connected subset $U$ of $X$ containing $a,b$", is reflexive. So if I can show that there is a path from a to a for any a in $X$, then the set ${a}$ can do the work.
The known definition of path from a point '$a$' to '$b$', by me is - There exists a continuous map '$f$' from $[0,1]$ to $X$ s.t. $f(0)=a$ and $f(1)=b$.
Define $f: [0,1] \to X$ by $f(t) = a$ for all $t \in [0,1]$. This is continuous for any $X$ because if $O \subseteq X$ is open, $f^{-1}[O] = \emptyset$ if $a \notin O$ and $f^{-1}[O] = [0,1]$ if $a \in O$, so the inverse image of an open set of $X$ is always open in $[0,1]$. (constant maps are always continuous between any pair of spaces).
So this forms a trivial path from a point to itself.