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The continuity of a function f from a topological space Z to a subspace Y of X is determined by checking whether the composition of f with the inclusion map i from Y to X is continuous.

Conditions: (X, $τ_X$) is a topological space.; Y is a subset of X with the subspace topology.; i: Y -> X is the natural inclusion map.; (Z, $τ_Z$) is any topological space.; f: Z -> Y is any function.

The continuity of a function f from a topological space Z to a subspace Y of X is determined by checking whether the composition of f with the inclusion map i from Y to X is continuous.

Conditions: (X, $τ_X$) is a topological space.; Y is a subset of X with the subspace topology.; i: Y -> X is the natural inclusion map.; (Z, $τ_Z$) is any topological space.; f: Z -> Y is any 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.

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.