Topology: continuity

Question

I am not sure how to approach the question.

Let $X=\{a,b\}$ and define two topological spaces on $X$ by

• $\tau_1=\{\emptyset, \{a\}, X\}$
• $\tau_2=\{\emptyset, \{b\}, X\}$

Is a function given by $f:(X, \tau_1) \rightarrow (X, \tau_2)$ defined by $f(x)=x$ continuous?

Should approach it by to cases $f(a)=a$, $f(b)=b$?

Answer 1

The only open sets in $\tau_2$ are $\emptyset$, $\{b\}$, and $X$. Notice that $f^{-1}(\{b\})=\{b\}$, which is not open in $\tau_1$, so $f$ is not continuous.