Topology Course lecture notes
Aisling McCluskey and Brian McMaster
In the study of metric spaces, we observed that:
(i) many of the concepts can be described purely in terms of open sets,
(ii) open-set descriptions are sometimes simpler than metric descriptions, e.g. continuity,
(iii) many results about these concepts can be proved using only the basic properties of open sets (namely, that both the empty set and the underlying set X are open, that the intersection of any two open sets is again open and that the union of arbitrarily many open sets is open). This prompts the question: How far would we get if we started with a collection of subsets possessing these above-mentioned properties and proceeded to define everything in terms of them?
(i) many of the concepts can be described purely in terms of open sets,
(ii) open-set descriptions are sometimes simpler than metric descriptions, e.g. continuity,
(iii) many results about these concepts can be proved using only the basic properties of open sets (namely, that both the empty set and the underlying set X are open, that the intersection of any two open sets is again open and that the union of arbitrarily many open sets is open). This prompts the question: How far would we get if we started with a collection of subsets possessing these above-mentioned properties and proceeded to define everything in terms of them?
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Εκδότης:
Aisling McCluskey and Brian McMaster
Γλώσσα:
english
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50
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PDF, 255 KB
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