Find comparable topologies on $X$ and $Y$ such that there exists $f:X⟶Y$ which is continuous or open with respect two but not the others.

Question

Let be $\mathcal T_X$,$\mathcal T'_X$ and $\mathcal T''_X$ three topologies on a set $X$ such that $$ \tag{1}\mathcal T'_X\subseteq\mathcal T_X\subseteq\mathcal T''_X $$ and analogously let be $\mathcal T_Y$, $\mathcal T'_Y$ and $\mathcal T''_Y$ three topologies on a set $Y$ such that $$ \tag{2}\mathcal T'_Y\subseteq\mathcal T_Y\subseteq\mathcal T''_Y $$ So I know that if $f$ is a function from $X$ to $Y$ continuous with respect $\mathcal T_X$ and $\mathcal T_Y$ then surely it is continuous with respect $\mathcal T''_X$ and $\mathcal T'_Y$ and analogously if it is open with respect $\mathcal T_X$ and $\mathcal T_Y$ then it is open with respect $\mathcal T'_X$ and $\mathcal T''_Y$ so that I ask to find these counterexamples which I feel exist.

• A function $f$ from $X$ to $Y$ continuous with respect $\mathcal T_X$ and $\mathcal T_Y$ but not continuous with respect $\mathcal T'_X$ and $\mathcal T''_Y$
• A function $f$ from $X$ to $Y$ open with respect $\mathcal T_X$ and $\mathcal T_Y$ but not open with respect $\mathcal T''_X$ and $\mathcal T'_Y$

So could some one help me, please?

Answer 1

Let's take $X = Y = \mathbb{R}$, with $T_X = T_Y$ being the standard topology of $\mathbb{R}$ $-$ whose basis is made of open intervals $(a,b) \subset \mathbb{R}$ for recall $-$, as well as the trivial topology $T_X' = T_Y' = \{\emptyset,\mathbb{R}\}$ and the discrete topology $T_X'' = T_Y'' = \mathcal{P}(\mathbb{R})$.

Then, a counter-example is simply given by the identity map. Indeed, one has for instance :

$\quad$a)$\;$ $\mathrm{id}^{-1}(\{0\}) = \{0\} \not\subset T_X'$ while $\{0\} \subset T_Y''$;

$\quad$b)$\;$ $\mathrm{id}(\{0\}) = \{0\} \not\subset T_Y'$ while $\{0\} \subset T_X''$.