trivial topology

Such a space is said to have the trivial topology.

A set with more than one element, with the trivial topology.

It has a non-trivial diffeology, but its D-topology is the trivial topology.

All subspaces of X have the trivial topology.

The coarsest topology on 'X' is the trivial topology.

See Trivial topology.

Two topological spaces carrying the trivial topology are homeomorphic iff they have the same cardinality.

In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space.

Slightly more generally, any space with a finite topology (only finitely many open sets) is compact; this includes in particular the trivial topology.

In some sense the opposite of the trivial topology is the discrete topology, in which every subset is open.

Despite its simplicity, a space X with more than one element and the trivial topology lacks a key desirable property: it is not a T space.

The "trouble" with the trivial topology is its poor separation properties: its Kolmogorov quotient is the one-point space.

Also, any set can be given the trivial topology (also called the indiscrete topology), in which only the empty set and the whole space are open.

The coarsest topology on X is the trivial topology; this topology only admits the null set and the whole space as open sets.

In the case of topologies, the greatest element is the discrete topology and the least element is the trivial topology.

The trivial topology belongs to a uniform space in which the whole cartesian product X x X is the only entourage.

The trivial topology (or indiscrete topology) on a set X consists of precisely the empty set and the entire space X.

Arbitrary products of trivial topological spaces, with either the product topology or box topology, have the trivial topology.

The trivial topology is the topology with the least possible number of open sets, since the definition of a topology requires these two sets to be open.

At the interface between an insulator with non-trivial topology, a so called topological insulator, and one with a trivial topology, the interface must become metallic.

G has a left adjoint F, creating the discrete space on a set Y, and a right adjoint H creating the trivial topology on Y.

In some ways, the opposite of the discrete topology is the trivial topology (also called the indiscrete topology), which has the fewest possible open sets (just the empty set and the space itself).

Examples are to be found when partitioning an interval (one refinement of being ), considering topologies (the standard topology in euclidean space being a refinement of the trivial topology).

Every non-empty subset of a set X equipped with the trivial topology is dense, and every topology for which every non-empty subset is dense must be trivial.

The property of separability does not in and of itself give any limitations on the cardinality of a topological space: any set endowed with the trivial topology is separable, as well as second countable, quasi-compact, and connected.