Topology of Mathematics

Define a topology on a nonempty set X as a collection of subsets closed under arbitrary unions and finite intersections, containing the empty set and X.

Examine when the four-set collection {∅, A, B, X} is a topology on X, showing it works if A and B partition X or if one contains the other.

This lecture demonstrates constructing four-member topologies on a set X via partitions into E and B, and shows they are ordered by inclusion.

Examine constructing topologies on the set X = {A, B, C} with exactly four members, using partitions of X by A and B and cases of total order by inclusion.

Explore discrete and indiscrete topologies on finite sets, defining discrete topology as the power set and comparing coarser and finer topologies by subset inclusion; illustrated with {a,b,c} and {1,2,3}.

The lecture demonstrates that a collection is a topology by closure under arbitrary unions and finite intersections, and that the intersection of two topologies is a topology.

Explain how the intersection of two topologies remains a topology, while the union of two topologies need not be a topology, with examples and key properties.

Explore the subspace topology: define subspace from a subset Y of a topological space X, with open sets as intersections of Y with X's open sets; illustrated with examples.

Demonstrate that the subspace of a discrete topological space is also discrete by analyzing the relative topology and open sets.

Show that the subspace of a discrete topology is also discrete, by verifying the relative topology on Y satisfies the topology axioms of unions, intersections, and the empty set.

Demonstrates that if y is a subspace of x with y ⊆ z ⊆ x, the subspace topology on y from x equals that from z.

Define open sets in a topology, compare subspaces, and prove that unions of open sets are open and finite intersections of open sets are open, with examples from discrete topology.

Explore closed sets in topology by defining them as complements of open sets and study examples, including discrete topology. Examine arbitrary intersections and finite unions of closed sets.

Define the closure of a subset as the intersection of all closed supersets, yielding the smallest closed superset containing the subset.

Explore a nondiscrete topology on X = {A,B,C,D,E} with open sets ∅, {A,B,C}, {D,E}, X; see complements yield closed sets matching open sets, so open and closed sets are identical.

Investigate closure of subsets in a topological space, including when a set is closed and equals its closure. Note that closure distributes over union but not necessarily over intersection.

Explore dense sets in a topological space, learn how closures and complements define open and closed sets, and study concrete examples in finite and discrete topologies.

Learn the interior point in a topological space: a point with an open neighborhood contained in a subset, illustrated with a simple example in X.