This book is about the branch of mathematics called topology. But its larger purpose is to illustrate how mathematics works: The interplay between intuition on the one hand and a pure mathematical formulation on the other. Thus, we develop the axioms for a topological space, formulate definitions within the context of those axioms and actually prove theorems from the axioms. But underlying all this is our intuition about topology. It is this intuition that guides and gives "meaning" to the definitions we make and to the theorems we prove. No prior knowledge of mathematics is assumed. In fact, these were originally the notes for a course for freshman non-scientists. This book, including over 100 figures and problem sets with solutions, should be of interest to those who would like to understand what mathematics is all about, as well as those who would like to learn about the this important branch of mathematics.