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)

• For a metric space (E, d), a Î E, and r Î R, the r-ball (neighborhood) with center a is Vd (a, r) = {x Î E : d(a, x) < r}. The deleted (or punctured) r-ball with center a is Vd' (a, r) = {xÎ E : 0 < d(a, x) < r}.

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.

• E is any set, T = P(E) (the discrete topology, the finest topology on E.)
• E is any set, T = {Æ , E} (the trivial, of indiscrete, topology on E.)
• E is any (non-null) set and p Î E, Tp = {Æ }È {X Î P(E) : p Î X} (a particular point topology) More to the *ahem* point, E = {0, 1}, T0 = {Æ , {0}, {0, 1}} (the Sherpinkski topology)
• E is any (non-null) set and p Î E, Tp = {E}È {X Î P(E) : p Ï X} (an exculded point topology)
• E is an infinite set, T = {X Î P(E) : E \ X is infinite} (the Fort topology) Here, for X Ì E, the complement of X in E (E\X) just means all the elements in E that are not in the subset X; than is, E\X = {x Î E : x Ï X}.
• Now, for a topology induced by a metric. Well, you had to know it was coming. ;-) Given a set E and a metric space (E, d), U Í E is "open relative to metric d" if, " x Î U, \$ rx Î R ' Vd (x, rx) = {y Î E : d(x, y) < rx } Í U. Then Td = {S Í E : S in open relative to d} is the topology induced by d.

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

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