By Iain T. Adamson

This ebook has been referred to as a Workbook to make it transparent from the beginning that it isn't a traditional textbook. traditional textbooks continue via giving in each one part or bankruptcy first the definitions of the phrases for use, the options they're to paintings with, then a few theorems related to those phrases (complete with proofs) and at last a few examples and routines to check the readers' realizing of the definitions and the theorems. Readers of this booklet will certainly locate all of the traditional constituents--definitions, theorems, proofs, examples and routines yet now not within the traditional association. within the first a part of the ebook might be came across a short evaluation of the elemental definitions of basic topology interspersed with a wide num ber of workouts, a few of that are additionally defined as theorems. (The use of the notice Theorem isn't meant as a sign of trouble yet of value and usability. ) The workouts are intentionally now not "graded"-after all of the difficulties we meet in mathematical "real lifestyles" don't are available order of trouble; a few of them are extremely simple illustrative examples; others are within the nature of instructional difficulties for a conven tional path, whereas others are rather tricky effects. No recommendations of the routines, no proofs of the theorems are incorporated within the first a part of the book-this is a Workbook and readers are invited to attempt their hand at fixing the issues and proving the theorems for themselves.

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The adherence of B, Adh B, is the set of its adherent points. Exercise 112. Show that Adh B = n XEB CI X. To do this prove that both sides of the equation are equal to the adherence of the filter generated by B . Exercise 113. Let (E, T) be a topological space, A a subset of E . Prove that a point x of E is adherent to A if and only if there is a filter F on E such that A E F and F converges to x. 36 Chapter 4 If x is adherent to A th en VT(x) U {A} gene rates a filter with the required property.

Exercise 80. Let (E ,T) be a separable topological space. Prove that if V is a T-open subset of E then (V ,Tv) is also separable. Exercise 81. Let E be an uncountable set, p a point of E and Tp the particular point topology on E determined by p. Let A = Cdp}. Prove that (E, T p ) is separable but that (A, (Tp)A) is not. Example 2. Let ((Ei ,Ti))iEI be a family of to pological spaces. Let E = 0 iEI E, and, for each index i in J, let 1I"i be the projection mapping from E to E i . The topology induced on E by the family of mappings (1I"i)iEI is called the product topology on E ; we denote it by 0 iEI i: The topological space (0 iEl e; 0 iElTi) is called the topological product of the family ((Ei ,Ti)) iEI.

Adamson, A General Topology Workbook © Birkhäuser Boston 1996 44 Chapter 5 To prove this result, show first that the canonical surjection TJ from E onto E / R is an open mapping (to do this let U be any T-open subset of E and prove that TJ+-(TJ-> (U)) = U) . Now let X = TJ(x) and Y = TJ(Y) be distinct points of E / R ; since X and Yare distinct, it follows that (x,y) f/. R and so Clr{x} :I Clr{y}; so there is a T-open set U containing one of x and y but not the other. Show that TJ->(U) contains the corresponding one of X and Y but not the other.