Point Set Topology

Point-Set Topology II

Math 6700–point Set Topology

Important Note: This document represents the notes I have taken during the Point Set Topology lectures of Dr. Vladimir Uspenskiy during the Fall 2013 term at Ohio University. As such, while I have done my best, I make no guarantees (or even implications) that everything contained therein is accurate or complete. While the material is obviously not my original work, this document is and as such ...

Institute for Mathematical Physics Nonstandard Analysis in Point{set Topology Nonstandard Analysis in Point-set Topology

We present Nonstandard Analysis by three axioms: the Extension, Transfer and Saturation Principles in the framework of the superstructure of a given innnite set. We also present several applications of this axiomatic approach to point-set topology. Some of the topological topics such as the Hewitt realcompactiication and the nonstandard characterization of the sober spaces seem to be new in the...

Notes on point set topology, Fall 2010

The proofs that these maps are continuous are simple estimates that you probably remember from calculus. Since the continuity of all the maps we’ll look at in these notes is proved by expressing them in terms of the maps a and m, we include the proofs of continuity of a and m for completeness. Proof. To prove that the addition map a is continuous, suppose x = (x1, x2) ∈ R and > 0 are given. We ...

Review of metric spaces and point - set topology

is not trivial to prove. In the one-dimensional case, the triangle inequality is an inequality on absolute values, and can be proven case-by-case. In R, it is best to use the following set-up. The usual inner product (or dot-product) on R is x · y = 〈x, y〉 = 〈(x1, . . . , xn), (y1, . . . , yn)〉 = x1y1 + . . .+ xnyn and |x| = 〈x, x〉. Context distinguishes the norm |x| of x ∈ R from the usual abs...