Properties of inclusion map between topological spaces. Let $X$ be a topological space and $Y$ a subset of $X$. Write $i: Y \to X$ for the inclusion map. Choose the correct statement:

*

*If $i$ is continuous, then $Y$ has the subspace topology.


*If $Y$ is an open subset of $X$, then $i(U)$ is open in $X$ for all subsets $U \subseteq Y$ that are open in the subspace topology on $Y$.
As-

*

*Since $i$ is continuous means $i^{-1}(V)$ is open in $Y$ for any open $V$ in $X$. So can we say first option is wrong as $Y$ can have any topology which is stronger than the subspace topology.


*Since $U$ is open in subspace topology which means there exist an open  set $V \in X$ such that $U = V \cap Y$.  Then $i(U) = i(V\cap Y) $
How to think further?

I came across a map like: (X, T1) and (X, T2) is a topological space with topologies T1 and T2 where T1 is stronger than T2. Can we say that a map from (X, T1) to (X, T2) is an inclusion map? Anybody please.
