S. D. V. Stephens

notes / Math 55b / chapter 1: Metric and Topological Spaces / From my notebook

N6 Bases, Subspaces, Products and Homeomorphisms

Basis

Definition N6.1 (Basis). A basis is a collection of opens such that satisfies

  1. for all there exists such that ;
  2. if and , then there exists such that .

Subspace topology

Definition N6.2 (Subspace topology). For , define

A rectangle X containing a curve Y; a dashed open set U of X meets the curve in a highlighted arc U intersect Y, an open set of the subspace topology.

Product topology

Definition N6.3 (Product topology). The product topology on (each with ) is generated by the basis

Two overlapping open rectangles U1 x V1 and U2 x V2 in X x Y with their intersection shaded, and next to them a disk covered by infinitely many small rectangles.

It takes infinitely many rectangles to cover a ball. The intersection of two basis elements is again one:

If and are metric spaces, we can build a distance whose metric topology is the product topology. Balls in = (balls in ) (balls in ).

We could also use These also define the product topology. Similarly for , a finite product.

Homeomorphisms

Definition N6.4 (Homeomorphism). A homeomorphism is an isomorphism of topological spaces: and are homeomorphic if there exist continuous bijections , with , .

Under a homeomorphism, open sets correspond to open sets: for and matched up by and ,

Remark. The inverse of a continuous bijection needn’t be continuous. Take with topologies ( strictly finer). Then is continuous, but not a homeomorphism.

Boundedness is not topological

Definition N6.5 (Bounded). A metric space is bounded if the diameter

This is not a topological property, and thus will not be invariant under homeomorphism.

Example N6.6. is a homeomorphism.

Graph of tan x on the open interval from -pi/2 to pi/2, with vertical asymptotes at both ends, showing a bounded interval homeomorphic to the whole real line.