Reference no: EM13990654

Given a product X = ITi ∈I Xi of topological spaces (Xi , T ), with product topology, and a directed set J , a net in X indexed by J is given by a doubly indexed family {x ji } j ∈ J,i ∈I . Show that such a net converges for the product topology if and only if for every i ∈ I , the net {x ji } j ∈ J converges in Xi for Ti . (For this reason, the product topology is sometimes called the topology of "pointwise convergence": for each j , we have a function i f→ xij on I , and convergence for the product topology is equivalent to convergence at each "point" i ∈ I . This situation comes up especially when the Xi are [copies of] the same space, such as R with its usual topology. Then ITi Xi is the set of all functions from I into R, often called RI .)