By John B. Conway

This textbook in aspect set topology is geared toward an upper-undergraduate viewers. Its mild speed should be necessary to scholars who're nonetheless studying to jot down proofs. necessities comprise calculus and at the very least one semester of research, the place the coed has been accurately uncovered to the information of uncomplicated set idea corresponding to subsets, unions, intersections, and features, in addition to convergence and different topological notions within the genuine line. Appendices are incorporated to bridge the space among this new fabric and fabric present in an research path. Metric areas are one of many extra familiar topological areas utilized in different parts and are for that reason brought within the first bankruptcy and emphasised during the textual content. This additionally conforms to the method of the booklet firstly the actual and paintings towards the extra normal. bankruptcy 2 defines and develops summary topological areas, with metric areas because the resource of thought, and with a spotlight on Hausdorff areas. the ultimate bankruptcy concentrates on non-stop real-valued capabilities, culminating in a improvement of paracompact areas.

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**Example text**

Proof. 9IJ = {Un} be a countable base of'iJ. 9IJ [where n(x) depends on x] such that x E Un(x) C P. 9IJ, {Un(x)} is a countable covering of X. Hence (X, 'iJ) is a Lindelof space. §12 • 33 Metric Spaces Proposition 15. In a second countable topological space (X uncountable subset A has a limit point in A. ~), every Proof. Suppose 93' = {Un} is a countable base of~. Suppose A does not have a limit point in A. Then for each a E A there exists Un(a) in {Un} such that a E Un(a) and Un(a) does not contain any other element of A.

Al > a 2 and UI C U2 ), we have p(al , UI ) > p(a2 , U 2 ). The range of pis cofinal in r because {x"" a E r} is frequently in each U. , {xp, {J E B} is eventually in U. Therefore {xp, {J E B} converges to x. Conversely, suppose a sub net of {x"" a E r} converges to x. If x is not a limit point of {x"" a E r} then there is a neighborhood U of x such that {x"" a E r} is not frequently in U. That means it is eventually in ~U. Hence a subnet of {x"" a E r} is in ~U and therefore it cannot converge to x.

Therefore at least one of r A , r E is cofinal with r. Hence at least one of the subsets {x", a E r A } and {x", a E r B } of {x", a E r} has the property that {(x"' a ErA), x} Ef(X) or ({x", a ErE), x) Ef(X) by condition (b). Hence x E q;(A) u q;(B) and we have shown that q;(A) u q;(B) = q;(A u B). Thus all the conditions of Theorem I are satisfied. Therefore there exists a unique topology Wi on X such that q;(A) = ClwA for each A E g(X). Finally, to show that lim"x" = x iff ({x,,}, x) E f(X), let ({x,,}, x) E f(X), and assume lim"x~ =I=- x.