Extended topology: Filters and convergence I

ConclusionThe above-mentioned results show some of the advantages of using extended filters. It has already been shown that the family of all proper extended filters forms a complete lattice. Many of the results obtained by using Cartan's filters can be obtained by using proper extended filters; it is not necessary to impose the condition that they should be closed with respect to finite intersections. And for most purposes it is sufficient if extended filters have the pair-wise intersection property; for instance limits are unique under this condition.In the next paper I propose to take up the question of convergence. Convergence will be dealt with in a setting more general than a topology of which convergence is extended topologies and topologies will be special cases. Wherever convenient relation theory will be used in dealing with extended topology which generalizes extended topology defined in terms of expansive functions.