Subspace topology

The topology on a subset of a topological space whose open sets are intersections of that subset with open sets of the ambient space.

Subspace topology theorem: A theorem describing how subspace topologies interact with ambient open and closed sets. It characterizes which ambient open or closed sets induce open or closed subsets.
Topological space: A set equipped with a collection of subsets designated as open and satisfying three axioms. The subspace construction starts with an ambient topological space.
Interval topology: The topology on an interval inherited from the real line’s usual topology. Intervals illustrate how ambient open sets induce neighborhoods at endpoints and interior points.
Discrete topology: The topology in which every subset of a set is open. An inherited topology need not make every subset open.
Subspace topology and continuity: The relationship between continuity on a subset and continuity inherited from an ambient space. The inherited topology makes the inclusion map continuous and gives a criterion for restricted maps.
Open set: A member of a topology, interpreted as an open region of the space. Intersecting ambient open sets with the subset produces its open sets.
Product topology: The topology on a product formed from open sets in its coordinate spaces. Subspaces of products inherit open sets from the product topology.
Indiscrete topology: The topology containing only the empty set and the entire underlying set. Subspace topologies can have many open sets, unlike this minimal topology.
Subspace topology and closure: The relationship between closure in a subspace and closure in its ambient space. Closure in the subset is obtained by intersecting the ambient closure with the subset.
Topology: A collection of subsets of a set that specifies which subsets count as open. A subspace topology is one way to construct a topology on a set.

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