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The treatment of the subject of this text is not encyclopedic, nor was it designed to be suitable as a reference manual for experts. Rather, it introduces the topics slowly in their historic manner, so that students are not overwhelmed by the ultimate achievements of several generations of mathematicians. Careful readers will see how topologists have gradually refined and extended the work of their predecessors and how most good ideas reach beyond what their originators envisioned. To encourage the development of topological intuition, the text is amply illustrated. Examples, too numerous to be completely covered in two semesters of lectures, make this text suitable for independent study and allow instructors the freedom to select what they will emphasize. The first eight chapters are suitable for a one-semester course in general topology. The entire text is suitable for a year-long undergraduate or graduate level curse, and provides a strong foundation for a subsequent algebraic topology course devoted to the higher homotopy groups, homology, and cohomology.