Point-set topology

Abstract

Remember that all of the examples and theorems in the previous chapter dealt with sets in ℝⁿ and inherited a lot of structure from the standard euclidean structure of ℝⁿ. In particular, the definition of our neighborhoods Dⁿ(x, r} = {y ∈ ℝⁿ : ∥y − x∥ < r makes use of the euclidean distance. It is possible to study point-set topology on a much more abstract level, by using different neighborhoods. Notice that all the definitions in Chapter 2 were based on the concept of a neighborhood of a point or on the concept of an open set. The definitions of interior, limit point, closed set, connectedness, and continuity all can be rewritten to depend only on the ideas of neighborhoods and open sets. We defined a neighborhood to be an open disc around a point, but this choice was really arbitrary.

“I preach mathematics; who will occupy himself with the study of mathematics will find it the best remedy against the lusts of the flesh.”

Thomas Mann