I will provide a brief introduction to general nonstandard analytic thinking, and then focus on how such thinking can help us in understanding some basic notions in point-set topology. 
 
Non-standard analysis is most famous for providing a rigorous way to do Calculus using infinitesimals. These infinitesimals are actual numbers in any "non-standard extension" of the real numbers. Intuitively, we imagine infinitesimals as the points we would see if we could "zoom in infinitely" at the origin of the number line. Similarly, in the context of topology, a non-standard extension of topological space is a typically much richer space --- and there is a sense in which we can "zoom in infinitely". Non-standard ideas make many concepts in point-set topology much more intuitive. As an example, we will see how Tychonoff's Theorem is almost immediate as soon as we understand what the non-standard way to think about compactness is.