Definition of a Topological Space and the Product of Two Topological Spaces
Definition of a Topological Space:
A topological space is a set X together with a collection of subsets of X, called open sets, that satisfy the following three axioms:
- The union of any collection of open sets is open.
- The intersection of any finite collection of open sets is open.
- The empty set and X itself are both open.
Definition of the Product of Two Topological Spaces:
Given two topological spaces X and Y, the product topological space X x Y is defined as the set X x Y together with the collection of subsets of X x Y that are the images of open sets in X and Y under the projection maps.
In other words, the product topology on X x Y is the coarsest topology that makes both projection maps continuous.
Related Questions and Answers
- What is the base of a topological space? Answer: A set of open sets whose union is the entire space.
- What is a closed set in a topological space? Answer: A complement of an open set.
- What is a continuous map between two topological spaces? Answer: A map that preserves the topological structure.
- What is the subspace topology? Answer: The topology on a subset of a topological space that is induced by the open sets of the original space.
- What is the product sigma-algebra? Answer: The sigma-algebra on the product space of two measurable spaces that is generated by the product sets.
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