By John B. Conway

This textbook in element set topology is aimed toward an upper-undergraduate viewers. Its mild velocity can be important 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 scholar has been safely uncovered to the tips of easy set concept resembling subsets, unions, intersections, and features, 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 direction. Metric areas are one of many extra universal topological areas utilized in different parts 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 e-book first of all the actual and paintings towards the extra basic. bankruptcy 2 defines and develops summary topological areas, with metric areas because the resource of idea, and with a spotlight on Hausdorff areas. the ultimate bankruptcy concentrates on non-stop real-valued services, culminating in a improvement of paracompact spaces.

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**Extra resources for A Course in Point Set Topology**

**Example text**

All topological spaces encountered in this book will be assumed to be Hausdorﬀ. Incidentally, another term for Hausdorﬀ is a T2 -space. The reason for this term is part of what are called separation axioms or properties. Yes, there is a T1 -space and even a T0 -space as well as T3 and T4 . It is a long story. We are going to avoid this terminology; it just strikes me as not conveying anything except, possibly, a hierarchical code. Nevertheless, the underlying concepts will be seen subsequently in Sects.

Continuing, we get that En ⊆ A for all n ≥ 1. That is, E = A, and so E is connected. 7. The union of two intersecting connected subsets of a metric space is connected. 8. If (X, d) is a metric space, then a component of X is a maximal connected subset of X. The word maximal in the deﬁnition means that there is no connected set that properly contains it. Thus, if C is a component of X and D is a connected subset of X with C ⊆ D, then D = C. A component is the correct interpretation of the word part used in the introduction of this section.

Xn be an -chain from x to y. By Exercise 8, there is an -chain from x to y in G whenever 0 < < . Applying this with 0 < < min{ , r}, we may assume < r. 14 we see that this implies there is an -chain in G from x to z. Thus z ∈ D, and so D is relatively closed in G. We note that (i) of the deﬁnition of an -chain was used to establish that the condition in part (b) was suﬃcient for connectedness. Without this, the result is false, as we see in the following example. 16. Let X = {(x, y) ∈ R2 : y > x−1 } ∪ {(x, y) ∈ R2 : y < 0}.