How is the continuity of a function into a subspace determined?

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.

Reasoning, step by step

1. Identify the function f from Z to Y.
2. Identify the inclusion map i from Y to X.
3. Form the composition i ∘ f from Z to X.
4. Check if i ∘ f is continuous.
5. Conclude that f is continuous if and only if i ∘ f is continuous.

Example

The handwritten text states: 'For every topological space (Z, $τ_Z$) and every function f: Z -> Y, f is cont iff the map if: Z -> X is cont.'

Common misconceptions

• Assuming that f is continuous if and only if it maps open sets in Y to open sets in Z.
• Ignoring the role of the inclusion map in determining continuity.

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