Comparison of Topologies - Properties

Properties

Let τ1 and τ2 be two topologies on a set X. Then the following statements are equivalent:

  • τ1 ⊆ τ2
  • the identity map idX : (X, τ2) → (X, τ1) is a continuous map.
  • the identity map idX : (X, τ1) → (X, τ2) is an open map (or, equivalently, a closed map)

Two immediate corollaries of this statement are

  • A continuous map f : X → Y remains continuous if the topology on Y becomes coarser or the topology on X finer.
  • An open (resp. closed) map f : X → Y remains open (resp. closed) if the topology on Y becomes finer or the topology on X coarser.

One can also compare topologies using neighborhood bases. Let τ1 and τ2 be two topologies on a set X and let Bi(x) be a local base for the topology τi at x ∈ X for i = 1,2. Then τ1 ⊆ τ2 if and only if for all x ∈ X, each open set U1 in B1(x) contains some open set U2 in B2(x). Intuitively, this makes sense: a finer topology should have smaller neighborhoods.

Read more about this topic:  Comparison Of Topologies

Famous quotes containing the word properties:

    “A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
    —Ralph Waldo Emerson (1803–1882)

    “The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
    —John Locke (1632–1704)