Abstract
This chapter marks a shift from a study of algebraic structures, and toward topological concepts that serve as the foundation for the geometric structures that feature prominently in the modern-day practice of mathematical physics. The goal of this chapter is to introduce the essential elements of general topology and the properties of topological spaces in a manner appropriate for a reader with no previous background in the subject. Chapters 7 and 8 build on these themes.
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Notes
- 1.
A different approach to topology is distinctly combinatorial in nature. Combinatorial methods are prominent in the theory of networks, for example, and they overlap with general topology in the field of algebraic topology where algebraic methods are used to explore topological properties of spaces. In many respects, algebraic topology is where the field of topology had its historical beginnings in the late ninteenth century (see [3]; also [7], Chap. 50).
- 2.
From the Latin: homeo-, meaning similar; versus homo-, the same. See also the comments at the end of Sect. 1.3.
- 3.
For a review, see Sect. 5.7.
- 4.
As an aside, we note that if \(\bar{A} = A\), then A is said to be everywhere dense in X. Alternatively, if the complement of the closure of A is everywhere dense, that is, if \(A = ({\bar{A}})^c\), then A is said to be nowhere dense in X.
- 5.
The important concepts of completeness, continuity and convergence—as well as compactness—are discussed in Sect. 6.5.
- 6.
See the definitions and discussion of norms, metrics and distance functions in Sect. 4.2.3.
- 7.
Terminological nuances vary among authors. Our convention is “regular = criterion (2)” so that “\(T_3 = T_1 + \mathrm {regular}\).” This follows the convention in [2, 6, 11]. Another convention is to say \(T_3 = \mathrm {criterion(2)}\), in which case “\(\mathrm {regular} = T_1 + T_3\).” This latter convention is followed in [12].
- 8.
- 9.
For a formal proof that a metric space is \(T_4\), see, for example, [11], pp. 30–31.
- 10.
See, for example, [11], Sect. 2.2.
- 11.
Our treatment of continuous functions in Sect. 1.3.2 was essentially expressed in terms of convergent sequences: if \(f(x_n)\) converges to f(x) as \(x_n\) converges to x, then f is continuous.
- 12.
- 13.
The term “limit point” tends to be used in the context of metric spaces, whereas “accumulation point” is used for sets. The term “cluster point” is discussed below.
- 14.
- 15.
If there were a distance function, we could think of D as the set of all distances from the set A for each point along the path. The binary relation \(\succeq \) in D would then give a partial ordering to those distances.
- 16.
For a review of Cartesian products of sets and projection maps, see Sect. 1.4.
- 17.
If \(X_1\) and \(X_2\) are multi-dimensional, the “coordinate” \(x_1\) as written here has multiple components incorporated within it, and similarly for \(x_2\). Hence, we use the plural, “coordinates.”
- 18.
We base our account here on the brief summary in [6], p. 88.
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Starkovich, S.P. (2021). Fundamental Concepts of General Topology. In: The Structures of Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-73449-7_6
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