OPEN AND CLOSED SETS . COMPLETE METRIC SPACES

This article modern topology within open and closed sets and their metric spaces in the structure main role strict and systematic learning, especially to the fullest accent to give presented Research metric and topological in spaces open and closed sets concepts from formalization begins, their interior, closure and border operators through equivalence emphasizes. Classic to definitions based on the article associations and intersections, borders points and openness, closedness and continuity between dependency such as main features is studied. Article main direction of all Cauchy sequences approach with described complete metric spaces is the concept of. At work in analysis completeness importance, especially of borders the existence and functional of processes stability in providing is studied . Complete of spaces structural wealth show Cantor intersection for theorem and Baire category theorem such as main results discussion Statistical and bibliometric analyses this shows that functional analysis and practical in mathematics modern more than 70% of the research completeness with related to arguments relies on and this his/her theoretical and practical in the fields importance emphasizes.