By John B. Conway

This textbook in aspect set topology is geared toward an upper-undergraduate viewers. Its mild speed could be worthwhile to scholars who're nonetheless studying to write down proofs. necessities contain calculus and no less than one semester of research, the place the scholar has been correctly uncovered to the guidelines of easy set idea similar to subsets, unions, intersections, and services, in addition to convergence and different topological notions within the actual line. Appendices are integrated to bridge the space among this new fabric and fabric present in an research path. Metric areas are one of many extra wide-spread topological areas utilized in different components and are accordingly brought within the first bankruptcy and emphasised through the textual content. This additionally conforms to the method of the ebook to begin with the actual and paintings towards the extra basic. bankruptcy 2 defines and develops summary topological areas, with metric areas because the resource of proposal, and with a spotlight on Hausdorff areas. the ultimate bankruptcy concentrates on non-stop real-valued features, culminating in a improvement of paracompact spaces.

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It is easy to check that f has the desired properties. 10. 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.

6) If (X, d) is a complete metric space and E ⊆ X, show that E is totally bounded if and only if cl E is compact. (7) (a) If G is an open set and K is a compact set with K ⊆ G, show that there is a δ > 0 such that {x : dist (x, K) < δ} ⊆ G. (b) Find an example of an open set G in a metric space X and a closed, noncompact subset F of G such that there is no δ > 0 with {x : dist (x, F ) < δ} ⊆ G. (8) If (X1 , d1 ), (X2 , d2 ) are metric spaces, show that X1 × X2 is compact if and only if both X1 and X2 are compact.

4 (Intermediate Value Theorem). If f : (X, d) → R is continuous, X is connected, a, b ∈ f (X) with a < b, then for any number c in the interval [a, b] there is a point x in X with f (x) = c. Proof. 2). Since a, b ∈ f (X), it must be that [a, b] ⊆ f (X). 5. (a) If x, y ∈ Rq , then the straight line segment [x, y] ≡ {ty + (1 − t)x : 0 ≤ t ≤ 1} is connected. In fact, t → ty + (1 − t)x is a continuous function from the unit interval into Rq . Since the unit interval is connected, so is its image under this continuous mapping.