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:
“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)
“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)