Abstract
Many examples play an important role in the study of general topology. This chapter presents some of the examples that are of general interests. ω denotes the first infinite cardinal and c = 2ω. The set of real numbers is denoted by ℝ, ℚ is the set of rational numbers, and ℕ is the set of positive integers. The real line ℝ is assumed to have the usual topology. The Alexandroff double circle is a classic example of a first-countable, compact, non-metrizable space that is easy to describe. To analyze it, two concentric circles (C1 and C2) are considered in the plane ℝ2 where points are represented by polar coordinates. The resulting space A is called the Alexandroff double circle. The space A is first-countable, compact, Hausdorff, and hereditarily normal, but is not perfectly normal because the open set C2 is not an Fσ –set of A. Moreover, A does not satisfy the countable chain condition because C2 is an uncountable set of isolated points. Thus, A is neither separable nor hereditarily LindelŠf.
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