By John B. Conway

ISBN-10: 3319023683

ISBN-13: 9783319023687

This textbook in element set topology is aimed toward an upper-undergraduate viewers. Its mild velocity should be invaluable to scholars who're nonetheless studying to write down proofs. must haves comprise calculus and a minimum of one semester of study, the place the scholar has been competently uncovered to the information of uncomplicated set thought similar to subsets, unions, intersections, and services, in addition to convergence and different topological notions within the genuine line. Appendices are incorporated to bridge the distance among this new fabric and fabric present in an research path. Metric areas are one of many extra generic 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 booklet first of all the actual and paintings towards the extra normal. 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 capabilities, culminating in a improvement of paracompact areas.

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**Extra info for A Course in Point Set Topology (Undergraduate Texts in Mathematics)**

**Sample text**

If (X, d) is a compact metric space and f : X → R is a continuous function, then there are points a and b in X such that f (a) ≤ f (x) ≤ f (b) for all x in X. Proof. 2 that f (X) is a closed and bounded subset of R. Put α = inf{f (x) : x ∈ X}, β = sup{f (x) : x ∈ X}. 3) of R; this proves the corollary. We need two more definitions. 4. Say that a subset K of the metric space (X, d) is totally bounded if for any radius r > 0 there are points x1 , . . , xn in K such that n K ⊆ k=1 B(xk ; r). A collection F of subsets of K has the finite intersection n property (FIP) if whenever F1 , .

Metric Spaces The last proposition is a way to combine continuous functions to obtain another continuous function. Here is another. Recall that if f : X → Z and g : Z → W are functions, then the composition of f and g is the function g ◦ f : X → W defined by g ◦ f (x) = g(f (x)). 8. 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.

In just 3 years he left his mark on mathematics. 3. Continuity 19 Recall that a map f : X → Z is injective if it is one-to-one; that is, if f (x) = f (y), then x = y. The function is surjective if it is onto; that is, for any z in Z there is a point x in X with f (x) = z. If f is both injective and surjective, then it is said to be bijective. When f is bijective, we can define the function f −1 : Z → X by letting f −1 (z) equal the unique point x in X such that f (x) = z. 7. I am afraid this is something we will have to live with.

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

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