Topology

 

There are various approaches to topology--point-set, algebraic, geometric. This page assumes some general, basic understanding of mathematics. Really, as long as you aren't actually math-phobic, it'll be fine.

Y Getting the Preliminaries out of the Way

(*) E = R, d(x, y) = |y - x| (the usual metric, a special case of the Euclidean metric below)

(*) E = Rn, d(x, y) = (the Euclidean metric) The notation of Rn, refers to the arrays, n-tuples, single column matrices of Euclidean n-space with elements or points often denoted like x = (x1, x2,... xn). Notice that, if n = 1, then this is just the usual metric.

(*) Given a set A, define E as the set of all Real-valued, bounded functions of A (E = {f : f :A® R, is bounded}) and d is defined by d(f, g) = supxÎ A|g(x) - f(x)| " f, g Î E. (the uniform metric)

(*) Given a set E, define a metric by . (the discrete metric)

Y Topological Spaces

Given a set E, a topology on E is T Í P(E) such that (i) E, Æ Î T, (ii) S1,...,Sn Î T Þ È i=1,...n Si Î T, and (ii) S1,...,Sn Î T Þ Ç i=1,...n Si Î T. [Note: The second condition does not require that the family of sets be finite, but the third condition does.} A topological space is an ordered pair, (E, T), where E is the "underlying set" and T is the topology on E. If T and T' are topologies on E and T Ê T', then T is finer than T' and T' is coarser than T.

A base for a topology T is a subset B Í T characterized by, " X Î T, $ Y1...Yn, Î B ' È i=1,...n Yi = X. The notion of a base is very nice so that we can talk about a topology without having to write out the whole topology. Suppose you have a set of sets where each set contains a spice; one set for parsley, one for anise, one for sassafras, one for rosemary, etc.... Now, we could mix'n'match our spices to make a topology--make various combinations, intersect/combine those combinations with the other combinations and single spices. Now matter what topology we make--or how big the mess gets in the kitchen--the base will be the collection (set) of singletons containing the original spices. In fact, let E be any set (such as a set of spices) and B be a set of subsets of E (such as a set of spice jars, even if a spice jar contains two or more spices). Suppose B satisfies two conditions: (i) È B = E and (ii) " S1, S2 Î B with x Î S1 Ç S2 Î B, $ S3 Î B ' x Î S3 Í S1 Ç S2. (In our example, a jar containing oregano and cilantro next to a jar of parsley and rosemary does not necessitate jars for all the separate spices because the intersection is null.) Then there is a unique topology TB on E for which B is a base.

Y Stuff to do in topological spaces

Y Links and Thinks

 

 

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