Glossary of Topology - I

I

Identification map

See Quotient map.

Identification space

See Quotient space.

Indiscrete space

See Trivial topology.

Infinite-dimensional topology

See Hilbert manifold and Q-manifolds, i.e. (generalized) manifolds modelled on the Hilbert space and on the Hilbert cube respectively.

Inner limiting set

A G_δ set.

Interior

The interior of a set is the largest open set contained in the original set. It is equal to the union of all open sets contained in it. An element of the interior of a set S is an interior point of S.

Interior point

See Interior.

Isolated point

A point x is an isolated point if the singleton {x} is open. More generally, if S is a subset of a space X, and if x is a point of S, then x is an isolated point of S if {x} is open in the subspace topology on S.

Isometric isomorphism

If M₁ and M₂ are metric spaces, an isometric isomorphism from M₁ to M₂ is a bijective isometry f : M₁ → M₂. The metric spaces are then said to be isometrically isomorphic. From the standpoint of metric space theory, isometrically isomorphic spaces are identical.

Isometry

If (M₁, d₁) and (M₂, d₂) are metric spaces, an isometry from M₁ to M₂ is a function f : M₁ → M₂ such that d₂(f(x), f(y)) = d₁(x, y) for all x, y in M₁. Every isometry is injective, although not every isometry is surjective.