238: Product of Closed Sets Is Closed in Product Topology|T.B.P.

Topics

About: topological space

The reader will have a description and a proof of the proposition that any product of closed sets (1 from each constituent topological space) is closed in the product topology.

For any product topological space,

T := T_{1} \times T_{2} \times . . . \times T_{n}

, and any closed sets,

C_{i} \subseteq T_{i}

C_{1} \times C_{2} \times . . . \times C_{n} \subseteq T

is closed on

T

C_{1} \times C_{2} \times . . . \times C_{n} = (T_{1} ∖ U_{1}) \times (T_{2} ∖ U_{2}) \times . . . \times (T_{n} ∖ U_{n})

where

U_{i} \subseteq T_{i}

is open on

T_{i}

(T_{1} ∖ U_{1}) \times (T_{2} ∖ U_{2}) \times . . . \times (T_{n} ∖ U_{n}) = T_{1} \times T_{2} \times . . . \times T_{n} ∖ ((U_{1} \times T_{2} \times . . . \times T_{n}) \cup (T_{1} \times U_{2} \times . . . \times T_{n}) \cup . . . \cup (T_{1} \times T_{2} \times . . . \times U_{n}))

T_{1} \times T_{2} \times . . . \times U_{i} \times . . . \times T_{n}

is open,

((U_{1} \times T_{2} \times . . . \times T_{n}) \cup (T_{1} \times U_{2} \times . . . \times T_{n}) \cup . . . \cup (T_{1} \times T_{2} \times . . . \times U_{n}))

is open.

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