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Compactness: Does every sequence in the set get really close to some point in the set really often? Related to the Extreme Value Theorem. Fréchet Introduced metric spaces in 1905. A metric space is a set on which a distance function is defined with the properties: The distance from x to y is the same as the distance from y to x . Distances are never negative, and are zero precisely when we’re measuring the dis- tance from a point to itself. The triangle inequality is satisfied. Example: Taxi-cab geometry. Hausdorff Introduced a topological space that consisted of a set and a collection of neighborhoods
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Unformatted text preview: satisfying certain properties that generalize the concept of open intervals. • Bourbaki ◦ Not a person, but a pseudonym for a bunch of French mathematicians working together. ◦ Introduced the most abstract modern de±nition of a topological space: A set X along with a collection of subsets of X called open sets that have the following properties: ⋆ The intersection of two open sets is an open set. ⋆ The union of any collection of open sets is an open set. ⋆ X and the empty set are both open sets....
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