Infinite Strings Generated by Insertions

The process of generation of strings over a finite alphabet by inserting characters at an arbitrary place in the string is considered. A topology on the set of strings is introduced, in which closed sets are defined to be sets of strings that are invariant with respect to the insertion of characters. An infinite insertion string is defined to be a set of infinite sequences of insertions ending in the same open sets. The number of infinite insertion strings is proved to be countable. It is proved also that there is a relationship between the countability of the completion of partially well-ordered sets by infinite elements and the fulfillment of an analogue of Dickson's and Higman's lemmas for them.