Other Properties
- The monomorphisms in Top are the injective continuous maps, the epimorphisms are the surjective continuous maps, and the isomorphisms are the homeomorphisms.
- The extremal monomorphisms are (up to isomorphism) the subspace embeddings. Every extremal monomorphism is regular.
- The extremal epimorphisms are (essentially) the quotient maps. Every extremal epimorphism is regular.
- There are no zero morphisms in Top, and in particular the category is not preadditive.
- Top is not cartesian closed (and therefore also not a topos) since it does not have exponential objects for all spaces.
Read more about this topic: Category Of Topological Spaces
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)