Neighborhoods and Interior Points

Topology · Axiom Academy

LESSON Neighborhoods and Interior Points Understanding how points relate to sets through open neighborhoods Step 1: What is a Neighborhood? Let be a topological space and let . A set is called a neighborhood of if there exists an open set such that . In other words, a neighborhood of doesn't have to be open itself, but it must contain an open set around . Think of it as having "breathing room" around the point. When the neighborhood is itself an open set containing , we call it an open neighborhood of . This is a special case where we can take . The interval is an open neighborhood of The interval is a neighborhood of , but not an open neighborhood Let be a subset of a topological space . A point is called an interior point of if is a neighborhood of . Equivalently, is an interior point of if there exists an open set such that . Step 4: Identifying Interior Points Any point is an interior point The endpoints and are NOT interior points For example, around we can fit the open interval Every open interval contains irrational numbers We cannot find an open set around any rational that stays entirely in Step 5: The Interior Operation The interior of a set , denoted or , is the set of all interior points of : Step 6: Properties of the Interior (interior is contained in the set) (interior of interior equals interior) (interior of union vs union of interiors)

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