By John B. Conway

ISBN-10: 3319023683

ISBN-13: 9783319023687

This textbook in element set topology is geared toward an upper-undergraduate viewers. Its mild velocity could be helpful to scholars who're nonetheless studying to put in writing proofs. must haves comprise calculus and at the least one semester of study, the place the coed has been safely uncovered to the information of uncomplicated set concept reminiscent of subsets, unions, intersections, and capabilities, 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 general topological areas utilized in different parts and are for this reason brought within the first bankruptcy and emphasised during the textual content. This additionally conforms to the strategy of the booklet firstly the actual and paintings towards the extra common. bankruptcy 2 defines and develops summary topological areas, with metric areas because the resource of concept, 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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If F is a closed subset of X and G is an open set containing F , then there is a continuous function f : X → R such that 0 ≤ f (x) ≤ 1 for all x in X, f (x) = 1 when x ∈ F , and f (x) = 0 when x ∈ / G. Proof. In Urysohn’s Lemma, take A to be the complement of G and B =F. 4Pavel Samuilovich Urysohn was born in 1898 in Odessa, Ukraine. He was awarded his habilitation in June 1921 from the University of Moscow, where he remained as an instructor. He began his work in analysis but switched to topology, in which he made several important contributions, especially in developing a theory of dimension.

Proof. If a ∈ A, then d(x, a) ≤ d(x, y) + d(y, a); thus, taking the infimum over all a in A we get dist (x, A) ≤ inf{d(x, y) + d(y, a) : a ∈ A} = d(x, y) + dist (y, A). Reversing the roles of x and y we have dist (y, A) ≤ d(x, y) + dist (x, A), whence we get the inequality. 6. If A is a nonempty subset of X, then f : X :→ R defined by f (x) = dist (x, A) is a continuous function. 2. 7. If (X, d) is a metric space and f and g are continuous functions from X into R, then f +g : X → R and f g : X → R are continuous, where (f + g)(x) = f (x) + g(x) and (f g)(x) = f (x)g(x) for all x in X.

The composition of two continuous functions is also continuous. Proof. If f : X → Z and g : Z → W , then for any subset G of W , then we have that (g ◦ f )−1 (G) = f −1 [g −1 (G)]. Thus, if G is an open subset of W , it follows that (g ◦ f )−1 (G) is open in X. Hence g ◦ f is continuous. Later in the book we will present several results on manufacturing continuous functions from a metric space into the real numbers. We have seen examples of such continuous functions on specific metric spaces, but we want to show the existence of continuous functions with specific properties on arbitrary ones.

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A Course in Point Set Topology (Undergraduate Texts in Mathematics) by John B. Conway


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