Properties
Every topological space is dense in itself. For a set X equipped with the discrete topology the whole space is the only dense set. 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.
Denseness is transitive: Given three subsets A, B and C of a topological space X with A ⊆ B ⊆ C such that A is dense in B and B is dense in C (in the respective subspace topology) then A is also dense in C.
The image of a dense subset under a surjective continuous function is again dense. The density of a topological space is a topological invariant.
A topological space with a connected dense subset is necessarily connected itself.
Continuous functions into Hausdorff spaces are determined by their values on dense subsets: if two continuous functions f, g : X → Y into a Hausdorff space Y agree on a dense subset of X then they agree on all of X.
Read more about this topic: Dense Set
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)