Properties
A subset of is closed/open if and only if its preimage under fi is closed/open in for each i ∈ I.
The final topology on X can be characterized by the following universal property: a function from to some space is continuous if and only if is continuous for each i ∈ I.
By the universal property of the disjoint union topology we know that given any family of continuous maps fi : Yi → X there is a unique continuous map
If the family of maps fi covers X (i.e. each x in X lies in the image of some fi) then the map f will be a quotient map if and only if X has the final topology determined by the maps fi.
Read more about this topic: Final Topology
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)