royster chapeter 1 notes
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Typed reading notes by Phil dated 2.1.05, written as a separate file to be printed and later merged with his other Royster notes. They summarize sections 1.1 to 1.6: the history of topology, set notation and operations, product sets, functions (injective, surjective, bijective, images and inverse images), equivalence relations and classes, and countable versus uncountable sets.
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Royster Chapter 1 PhL 2.1.05
At some point I will merge these short notes into the others, but for now separate so I can print and add.
1.1 History. In 1676 Leibniz thought about a geometry of position. Then in 1700's Euler and Gauss got on board. Position or location meaning points on some kind of surface (space). In 1847 Listing used the actual word topology in a book, but the accepted name at that time was analysis situs (of location). But it was Bernard Riemann (1826-1866) who thought about connectivity and holes and about spaces beyond R3.
1.2 Sets and Set Operations The roster method is { 1,2,3}, the set builder method is { x R | x < 5 }. Synonyms are element, object, member. Convention is sets are caps, elements are lower case. When you say A B , what you really means is A B because the possibility of A = B is included. Intersection, union, etc.
A universal set is your frame of reference like "the reals".
Complement of A is written in several ways: CA = Ac = A'.
The notation A \ B means what is in A but not in B, so this is not complement "as such". This is really what you might mean by A - B. He calls A \ B by the name set difference. "A not in B". I think the complement follows, but he does not comment on this.
1.3 Product sets and theorems about such are stated.
1.4 Functions. When you write y = f(x), x is the preimage and y the image. The set of all such points are domain and range. (codomain = range). But then we re-use words so that f(A) = image for some set A in domain, and similarly f-1(B) would be the inverse image of B. One to one is injective, onto is surjective, and if you have both, you have a bijection and the inverse of f then exists. The identity and notion of a composite function are given. Def of a sequence.
1.5 Equivalence Relations. A relation for a set X has elements { x1, x2 } and is thus an element of X x X, a direct product set as we just learned above in 1.3 above. xRy means "x is related to y" in some way. In equivalence relation is a special case where we have xRx for all x, xRyyRx and finally xRy,yRz xRz. This special relation has the notation x ~ y. The equivalence class denoted [x] of x is the set of all equivalent y and of course includes x.
1.6 Cardinality. Denumerable = countably infinite = countable. Else uncountable. Rationals can be counted, reals cannot.