The ideas and methods of set theory and topology permeate modern mathematics. It is no wonder then that the elements of these two mathematical disciplines are now an indispensable part of basic mathematical training. Concepts such as the union and intersection of sets, countability, closed set, metric space, and homeomorphic mapping are now classical notions in the whole framework of mathematics.
The purpose of the present volume is to give an accessible presentation of the fundamental concepts of set theory and topology; special emphasis being placed on presenting the material from the viewpoint of its applicability to analysis, geometry, and other branches of mathematics such as probability theory and algebra. Consequently, results important for set theory and topology but not having close connection with other branches of mathematics, are given a minor role or are omitted entirely. Such topics are, for instance, axiomatic investigations, the theory of alephs, and the theory of curves.
Here is Kuratowski's Introduction to the Set Theory part of the text.
Here is Kuratowski's Introduction to the Topology part of the text.