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:
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“...there are important considerations in the world beyond plain sewing and teaching dull little boys the alphabet. Any woman who has brains and willing hands finds twenty remunerative occupations open to her where formerly she would have found merely the inevitable two—plain sewing, or the dull little boys. All she had to do is to make her choice and then buckle on her armor of perseverance, while the world applauds.”
—Clara (Marquise)
“I think middle-age is the best time, if we can escape the fatty degeneration of the conscience which often sets in at about fifty.”
—W.R. (William Ralph)