What is the universal property for subspace topology?

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.

Reasoning, step by step

1. Identify the topological space X and its subset Y.
2. Consider the natural inclusion map i from Y to X.
3. Take any topological space Z and a function f from Z to Y.
4. Check if f is continuous.
5. Check if the composition i ∘ f from Z to X is continuous.
6. Verify that f is continuous if and only if i ∘ f is continuous.

Example

The formula is written as $f \text{ is cont. } \iff i \circ f \text{ is cont.}$

Common misconceptions

• Assuming that continuity in the subspace topology is independent of the ambient space.
• Confusing the inclusion map with other types of maps.

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Explore next

• How is the continuity of a function into a subspace determined?
• What is the role of the inclusion map in the universal property of subspace topology?