Allan J. Sieradski, “An Introduction to Topology and Homotopy“,

Introduction

Geometry uses the notions of distance, length, area and volume to express certain properties of objects in space. There are, however, features of objects that are not based upon rigid measurements.

For example, when a spherical balloon is inflated, it may lose the geometric property that all its points are equidistant from a central point in space. But all its deformed versions, whether ellipsoid?like or whatever, retain the feature that they divide space into a bounded region and an unbounded region, and that it is impossible to travel within space from one region to the other region without passing thuough the deformed sphere. In addition, if the balloon is punctured by the removal of a single point, it ceases to divide space into two regions.

Stretching and shrinking objects are examples of continuous functions. Puncturing and tearing objects are examples of discontinuous functions. These distinctions can be made only when the objects under consideration are sets of points in which it is known which subsets are near which points. The fundamental requirement for continuity is that the function respect all the nearness relationships between subsets and points of the object.

The features of the object that are preserved by continuous functions with continuous inverses are called topological properties. There are enough such features to warrant their own study in the discipline of topology.

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