By John B. Conway

This textbook in element set topology is geared toward an upper-undergraduate viewers. Its mild velocity could be worthwhile to scholars who're nonetheless studying to put in writing proofs. must haves contain calculus and not less than one semester of research, the place the scholar has been competently uncovered to the tips of simple set concept resembling subsets, unions, intersections, and features, 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 path. Metric areas are one of many extra frequent topological areas utilized in different components and are accordingly brought within the first bankruptcy and emphasised during the textual content. This additionally conforms to the technique 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 notion, 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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**Example text**

1. A topological space is a pair of objects (X, T ), where X is a set and T is a collection of subsets of X satisfying the following conditions: (a) ∅, X ∈ T ; (b) if {Gi : i ∈ I} ⊆ T , then (c) if G1 , . . , Gn ∈ T , then i∈I Gi n k=1 Gk ∈T; ∈T. The collection T is called the topology on X and sets in T are called open sets. 2. 1, then (X, T ) is a topological space. (b) If X is any set and T = 2X , the collection of all subsets of X, then (X, T ) is a topological space. This topology is called the discrete topology on X.

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 .

We won’t see much of this topology. (f) If (X, T ) is a topological space, Y ⊆ X, and TY ≡ {Y ∩ G : G ∈ T }, then (Y, TY ) is a topological space. TY is called the subspace topology or relative topology deﬁned by T on Y . We note that this is consistent with what we did when discussing subspaces of a metric space. That is, if (X, d) is a metric space, T denotes the open sets in X, and Y ⊆ X, then TY is precisely the set of open subsets of Y obtained by restricting the metric d to Y . 8. 3. (X, T ) is called a Hausdorﬀ 1 space provided for any pair of distinct points x, y in X where there are disjoint open sets U, V such that x ∈ U and y ∈ V .