Abstract
Remember that all of the examples and theorems in the previous chapter dealt with sets in ℝn and inherited a lot of structure from the standard euclidean structure of ℝn. In particular, the definition of our neighborhoods Dn(x, r} = {y ∈ ℝn : ∥y − x∥ < r makes use of the euclidean distance. It is possible to study point-set topology on a much more abstract level, by using different neighborhoods. Notice that all the definitions in Chapter 2 were based on the concept of a neighborhood of a point or on the concept of an open set. The definitions of interior, limit point, closed set, connectedness, and continuity all can be rewritten to depend only on the ideas of neighborhoods and open sets. We defined a neighborhood to be an open disc around a point, but this choice was really arbitrary.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
“I preach mathematics; who will occupy himself with the study of mathematics will find it the best remedy against the lusts of the flesh.”
Thomas Mann
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer Science+Business Media New York
About this chapter
Cite this chapter
Kinsey, L.C. (1993). Point-set topology. In: Topology of Surfaces. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0899-0_3
Download citation
DOI: https://doi.org/10.1007/978-1-4612-0899-0_3
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-6939-7
Online ISBN: 978-1-4612-0899-0
eBook Packages: Springer Book Archive