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topology definition question
Posted:
Nov 16, 2012 4:42 PM
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It seems the most "common" definition of a topology is that T is a topology on a set X if
1. empty set is in T and
2. X is in T and
3. A and B are in T ==> A intersection B is in T and
4. {A_i} in T ==> Union A_i is in T.
But some authors imply that only 3 and 4 are necessary for the definition of a topology. For example, Kelley ("General Topology", 1955, page 37) only uses 3 and 4 and says that these imply X is in T. McCarty ("Topology...", page 87) says 1 and 2 are "completely unneeded".
My question is, is it really possible to exclude 1 and 2 from the definition such that 3 and 4 alone imply 1 and 2?
Suppose X:={x,y,z} and T:={ {x},{y},{x,y} }.
Then T satisfies conditions 3 and 4, but yet X is not in T.
So how is it possible to exclude 3 from the definition of a topology?
Many thanks in advance,
Dan
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