A space has the shrinking property if, for every open cover {V_a | a ∈ A}, there is an open cover {W_a | a ∈ A} with for each a ∈… - Mary Ellen Rudin

Geometric topology was really the dominant new topological theme in the 1950's and differential topology in the 1960's. Algebraic topology did not take a back seat in either development. But something happened in the 1960's which had profound effect upon the part of topology we are concerned with.
... Paul Cohen proved that it is consistent with the usual axioms for set theory that the continuum hypothesis be false.
In itself this theorem has few consequence in topology for there is very little one can do with not-CH alone. But the technique of proof, called forcing, has translations into Boolean algebra terms, into partial order terms, into terms which lead to remarkable combinatorial statements which are applicable to a wide variety of topological problems related to abstract spaces.

Souslin's conjecture sounds simple. Anyone who understands the meaning of countable and uncountable can "work" on it. It is in fact very tricky. There are standard patterns one builds. There are standard errors in judgement one makes. And there are standard not-quite-counter-examples which almost everyone who looks at the problem happens upon. S. Tennenbaum and others have shown that that it is consistent with the axioms of Zermelo-Fraenkel set theory that Souslin's conjecture be either true or false.

The purpose of this paper is to construct (without using any set theoretic conditions beyond the axiom of choice) a normal Hausforff space <math>X</math> whose Cartesian product with the closed unit interval <math>I</math> is not normal. Such a space is often called a Dowker space. The question of the existence of such a space is an old and natural one ...