By John B. Conway

This textbook in aspect set topology is geared toward an upper-undergraduate viewers. Its light speed might be important to scholars who're nonetheless studying to put in writing proofs. must haves comprise calculus and at the very least one semester of study, the place the coed has been effectively uncovered to the information of uncomplicated set concept comparable to subsets, unions, intersections, and capabilities, in addition to convergence and different topological notions within the genuine line. Appendices are integrated to bridge the space among this new fabric and fabric present in an research direction. Metric areas are one of many extra established topological areas utilized in different components and are for this reason brought within the first bankruptcy and emphasised in the course of the textual content. This additionally conforms to the technique of the publication 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 concept, and with a spotlight on Hausdorff areas. the ultimate bankruptcy concentrates on non-stop real-valued capabilities, culminating in a improvement of paracompact spaces.

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11. A compact metric space is separable. very active in the French government, serving in the French Chamber of Deputies (1924–1936) and as Minister of the Navy (1925–1940). He died in 1956 in Paris. 5. Connectedness 29 Proof. For each natural number n we can ﬁnd a ﬁnite set Fn such that X = {B(x; n−1 ) : x ∈ Fn }. Put F = ∞ n=1 Fn ; we will show that this countable set F is dense in X. In fact, if x0 is an arbitrary point in X and > 0, then choose n such that n−1 < . Thus, there is a point x in Fn ⊆ F with d(x0 , x) < n−1 < , proving that x0 ∈ cl F .

What are the deﬁnitions? I won’t tell you. If you are curious, you can look them up, but it will not be helpful. I will say, however, that using this terminology the Baire Category Theorem says that a complete metric space is of the second category. My objection stems from the fact that this “category” terminology does not convey any sense of what the concept is. There is another pair of terms that is used: meager or thin and comeager or thick. These at least convey some sense of what the terms mean.

For each natural number n we can ﬁnd a ﬁnite set Fn such that X = {B(x; n−1 ) : x ∈ Fn }. Put F = ∞ n=1 Fn ; we will show that this countable set F is dense in X. In fact, if x0 is an arbitrary point in X and > 0, then choose n such that n−1 < . Thus, there is a point x in Fn ⊆ F with d(x0 , x) < n−1 < , proving that x0 ∈ cl F . Exercises (1) Show that the union of a ﬁnite number of compact sets is compact. (2) If K is a subset of (X, d), show that K is compact if and only if every cover of K by relatively open subsets of K has a ﬁnite subcover.