Sierpinski Space
Sierpinski Space M. VELRAJAN 1. What is the smallest topological space which is neither indiscrete nor discrete? To answer this question we start with a small topological space. So we start with a set with only one point. Consider the set X consisting of only one point, say {0}. The discrete topology in X = {0} is same as the indiscrete topology {∅, X}. And it is the only possible topology in X. So, next we consider a set with 2 points. Consider the set X consisting of two points, say {0, 1}. Besides the discrete topology {∅, X, {0}, {1}} and indiscrete topology {∅, X} the only possible topologies in X are {∅, X, {0}} and {∅, X, {1}}. It is clear that the topological spaces X with {∅, X, {0}} and X with {∅, X, {1}} are homeomorphic under the homeomorphism of interchanging of 0 and 1. So, as two finite homeomorphic spaces have the equal number of open sets, there are only 3 non homeomorphic topologies in X = {0, 1}, name...