Definition Via Open Sets
Objects: all topological spaces, i.e., all pairs (X,T) of set X together with a collection T of subsets of X satisfying:
- The empty set and X are in T.
- The union of any collection of sets in T is also in T.
- The intersection of any pair of sets in T is also in T.
- The sets in T are the open sets.
Morphisms: all ordinary continuous functions, i.e. all functions such that the inverse image of every open set is open.
Comments: This is the ordinary category of topological spaces.
Read more about this topic: Characterizations Of The Category Of Topological Spaces
Famous quotes containing the words definition, open and/or sets:
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)
“We were young, we were merry, we were very very wise,
And the door stood open at our feast,
When there passed us a woman with the West in her eyes,
And a man with his back to the East.”
—Mary Elizabeth Coleridge (1861–1907)
“Until, accustomed to disappointments, you can let yourself rule and be ruled by these strings or emanations that connect everything together, you haven’t fully exorcised the demon of doubt that sets you in motion like a rocking horse that cannot stop rocking.”
—John Ashbery (b. 1927)