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Topology: An Inquiry-Based Approach

Steven Schlicker

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Section Summary

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Important ideas that we discussed in this section include the following.
  • πŸ”—
    A topology on a set X is a collection of open subsets of X. More specifically, a set Ο„ of subsets of a set X is a topology on X if
    1. πŸ”—
      X and βˆ… belong to Ο„,
    2. πŸ”—
      any union of sets in Ο„ is a set in Ο„, and
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      any finite intersection of sets in Ο„ is a set in Ο„.
    A topological space is a set along with a topology on the set.
  • πŸ”—
    Any arbitrary union of open sets is open and any finite intersection of open sets is open in a topological space.
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    It can be difficult to completely describe the open sets in a topology, and it can be difficult to work with arbitrary open sets. If a collection of simpler sets generate a topology, that collection of simpler sets is a basis for the topology. More formally set B is a basis for a topology on a set X if
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      For each x∈X, there is a set in B that contains x.
    2. πŸ”—
      If x∈X is an element of B1∩B2 for some B1,B2∈B, then there is a set B3∈B such that x∈B3βŠ†B1∩B2.
  • πŸ”—
    A subset A of a topological space X is a neighborhood of a point a∈A if there is an open set O contained in A such that a∈O.
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    A point x in a subset A of a topological space X is an interior point of A if A is a neighborhood of x. The interior of set A is the collection of all interior points of A.
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    A subset A of a topological space X is open if and only if A is equal to its interior.
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