Assume is first countable. Then, for each lies in , there is a countable local basis at . Let a countable union of countable product set be such a countable local basis at , where is a local basis at each for finitely many values of and for all other values of i. It follows that should be all countable since has a countable local basis for each x lies in by hypothesis. Thus, each can be the required countable local basis at each and we conclude that is first-countable. Since a countable union of is countable, for all but countable values of i. Thus, should be indiscrete for all but countable values of i.
Assume each is first-countable, and every space , except for a countable quantity, are indiscrete.
Let lies in . If is a countable local basis at for the space , then the countable union of products , where for finitely many values of and for all the other values of is the desired countable local basis for lies in . Thus, is first countable.