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Proof That a Continuous Function With Codomain Rn on a Closed Subset of a T4 Space X Has a Continuous Extension to X

Theorem

Ifis T4 andis a closed subset ofthen a continuous function

can be extended to a continuous function

Proof

is a continuous function ofmapping any element ofto an element of

Hence it can be written as

Each of the functionsis continuous and mapsinto

Tietze's extension theorem tells us that each of the functions gives rise to a continuous functionfrominto

Define

Thenis continuous and is a continuous extension offromto