Linear Order, Order Topology, Well Ordering - Slides | MTH 631, Study notes of Topology

Download Linear Order, Order Topology, Well Ordering - Slides | MTH 631 and more Study notes Topology in PDF only on Docsity! Assignment Read Section 14, Skim sections 9 and 10 Continue to work on homework Mth 631 – Fall 2008 Order Topology 1/8 Countable Dense sets Def. A subset D of a topological space X is dense in X if each nonempty open set in X contains a point of D. Examples: Theorem: 1 If X has a countable basis, then X has a countable dense subset. 2 If X is metrizable and has a countable dense subset, then X has a countable basis. Cor. R is not metrizable. Mth 631 – Fall 2008 Order Topology 2/8 Linear Order Def. A simple or linear order on X is a transitive relation < on X such that ∀ pair x , y in X , exactly one of the following holds: x < y , y < x , x = y . Mth 631 – Fall 2008 Order Topology 3/8 Order Topology Def. If X is linearly ordered with more than one element, the order topology on X is the topology with basis consisting of all intervals of one of the forms: (a,b) for a < b, [m,b) where m is the minimum element of X (if one exists) (a,M] where M is the maximum element of X (if one exists) Check: this collection of intervals is a basis for some topology, using the results of Section 13. Mth 631 – Fall 2008 Order Topology 4/8