What would you like to understand?

The subspace topology on a subset Y of a topological space X is characterized by the property that a function from any topological space Z to Y is continuous if and only if its composition with the inclusion map into X is continuous.

Conditions: (X, $T_X$) is a topological space.; Y is a subset of X.; i: $Y \hookrightarrow X$ is the natural inclusion.; (Z, $T_Z$) is any topological space.; f: $Z \to Y$ is a function.

The subspace topology on a subset Y of a topological space X is characterized by the property that a function from any topological space Z to Y is continuous if and only if its composition with the inclusion map into X is continuous.

Conditions: (X, $T_X$) is a topological space.; Y is a subset of X.; i: $Y \hookrightarrow X$ is the natural inclusion.; (Z, $T_Z$) is any topological space.; f: $Z \to Y$ is a function.