Abstract
Metric spaces are convenient for many applications in geometry and physics, but sometimes we need a broader framework: that of topological spaces. In this chapter we give a short introduction to this theory.
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Notes
- 1.
In the last, optional section of this chapter we introduce a generalization of sequences that is well suited for all topological spaces.
- 2.
- 3.
We do not exclude equality.
- 4.
- 5.
This is consistent with our terminology on metric spaces: the topology of a metric subspace coincides with the subspace topology.
- 6.
The topology of a product of metric spaces coincides with the product of the corresponding topologies. We will also define the product of infinitely many spaces in Sect. 2.4, p. 53.
- 7.
This definition is consistent with the definition of density in metric spaces; see pp. 18 and 42.
- 8.
All sets are closed in the discrete topology.
- 9.
See Sect. 2.5 for such a modification.
- 10.
By Proposition 2.18 (a) the hypotheses are satisfied if (Fn) is a non-increasing sequence of non-emptyclosedsets in acompacttopological space.
- 11.
- 12.
More precisely, but less intuitively, we could consider the functions x: I → X defined by the formula x(i): = x i .
- 13.
This notion is equivalent to the more elegant but less transparent notion of a filter. See Császár [118].
- 14.
Use the last property in the definition of directed sets.
- 15.
Adapt the solution of Exercise 1.7, pp. 32 and 301.
- 16.
This a generalization of the definition of the complex sphere in complex analysis.
Bibliography
Á. Császár, Foundations of General Topology, International series of monographs on pure and applied mathematics v. 35, Pergamon Press, New York, 1963.
A. Hajnal, P. Hamburger, Set Theory, Cambridge Univ. Press, 1999.
J. Kelley, General Topology, Van Nostrand, New York, 1954.
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Komornik, V. (2017). Topological Spaces. In: Topology, Calculus and Approximation. Springer Undergraduate Mathematics Series. Springer, London. https://doi.org/10.1007/978-1-4471-7316-8_2
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DOI: https://doi.org/10.1007/978-1-4471-7316-8_2
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