Let $p:( [0,1],T_1) \rightarrow ( \{0,1\},T_2)$ be the quotient map arising from the characteristic function on $[\frac{1}{2},1],$ where $T_1$ is the subspace topology induced on $[0,1]$ from the standard topology on $\mathbb{R}.$
Show the following claims about the quotient map p.
- Prove p is an open map. If not, give a counterexample.
- Prove p is a closed map. If not, give a counterexample.
Attempt:
The open sets in $T_1$ are of the form
(i) $(a,1],$ where $0 \leq a <1,$
(ii) $[0,b),$ where $0<b \leq 1,$
(iii) $(a,b),$ where $0 \leq a \leq b \leq 1,$
(iv) $\phi$ and the whole space $[0,1].$
Given that p is the characteristic function on $[\frac{1}{2},1].$ So, $$p(x)=\chi_{[\frac{1}{2},1]}= \begin{cases} 1, & x \in [\frac{1}{2},1] \\ 0, & x \in [0,\frac{1}{2}) \\ \end{cases}$$
I think the quotient topology induced on the set $\{ 0,1 \}$ by the quotient map $p$ is discrete topology. But, I am not getting any further ideas on how to go ahead in this problem. Any help would be highly appreciated.