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Springer International Publishing Switzerland 2014

Marcelo Epstein Differential Geometry Mathematical Engineering 10.1007/978-3-319-06920-3_1

1. Topological Constructs

Marcelo Epstein 1

(1)

Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, AB, Canada

1.1 Topological Spaces

1.1.1 Definition

Physical theory has imposed on us the need for the notion of a continuum . Indeed, more than two millennia ago, the great magister said that space is infinitely divisible into parts, themselves infinitely divisible. He also established that space has three dimensions, by which all bodies are bounded. Whatever else may be wrong in Aristotle, at least when it comes to this basic notion he got it right. Fast-forward to the early twentieth century to find mathematicians (Riesz, Hausdorff, Kuratowski) tackling and solving the following question: What is the minimal structure that a set must possess to sustain the notions of nearness and continuity ? The answer is, of course, a topology . The key is provided by the identification of certain subsets of as being open . This identification must satisfy three conditions, which are ultimately abstracted from the common notion of open set in :

The null set and the total set are open.

Any arbitrary union of open subsets is open.

The intersection of a finite number of open subsets is open.

The set is then said to have been endowed with a topology or to be a topological space .

1.1.2 Nearness and Continuity

A neighbourhood of a point is any open subset such that . Although we have not employed any metric notion to discriminate whether a neighbourhood is small or large, it is clear that if a neighbourhood happens to be a subset of another neighbourhood , then it is legitimate to conceive of as smaller than . Thus it becomes possible to express the idea of getting closer and closer to a point by a sequence of nested neighbourhoods, which is all we need to understand the notion of continuity.

A function between two topological spaces, and , is continuous at if, for every neighbourhood of q = f ( p ), there exists a neighbourhood of p such that . Notice how cleverly this definition emulates the classical calculus definition without using any metric concepts.

A function is continuous if it is continuous at every point p of its domain . Equivalently, a function is continuous if inverse images of open subsets in are open in .

Two topological spaces, and , are said to be homeomorphic , if there exists a continuous bijection (called a homeomorphism ) whose inverse is continuous. One of the main objectives of Topology is the study of properties that are preserved under homeomorphisms of topological spaces.

1.1.3 Some Terminology

A subset of a topological space is closed if its complement is open. The concepts of open and closed are not mutually exclusive. A subset may be simultaneously open and closed (such as is the case of the whole space) or be neither open nor closed.

A topological space is connected if it cannot be expressed as the union of two non-empty disjoint open subsets. Two important properties of a connected space are: (i) A topological space is connected if, and only if, the only subsets that are both open and closed are the empty set and the total space. (ii) Connectedness is preserved by continuous maps.

An open cover of a topological space is a collection U of open subsets whose union is . An open cover B is said to be a basis for if every open subset can be expressed as a union of sets in B . We also say that B generates the topology . A desirable property of a topological space is the so-called second-countability. A topological space is said to be second-countable if it has a countable basis.

Another desirable property that a topological space may enjoy is the following: given any two distinct points, a and b , there exist respective neighbourhoods, and , that are mutually disjoint, i.e., . If this happens to be the case,

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