Is $X$ a $T_1$ space? (i.e., given a pair of distinct points in $X$, does each one of them have a neighborhood not containing the other?)

Question

Define a topology $T$ on the closed interval $X = [−1, 1]$ by declaring a set to be open if it either does not contain the point $0$, or it does contain $(−1, 1)$.

1. Is $X$ a $T_1$ space? (i.e., given a pair of distinct points in $X$, does each one of them have a neighborhood not containing the other?)

2. Show that $X$ locally path-connected

Answer 1

If a space is T1 it is equivalent to saying that singleton sets are closed. You need to only verify whether singletons are closed or not.