- University of Wisconsin - Madison
- Mathematics 521
- Analysis Final
Last Modified: 2016-12-22
1) (+ closure), + commutes, + associative, + identity, + inverse
2) multiplicative closure, commute, associative, identity, inverse, distributivity
1) for any x,y in S, exactly one of the following holds: x=y, x<y, y<x
2) Transitivity: for any x, y, z in S, if x < y and
y<z, then x <z
1) if y is less than or equal to z, then x + y is less than or equal to x +z
2) if x is greater than or equal to 0 and y is greater than or equal to 0, then x*y is greater than or equal to 0.
i) (positivity) d(x,y) is greater than or equal to 0 and d(x,y) = 0 if and only if x = y
ii) (symmetry) d(x,y) = d(y,x)
iii) (Triangle Inequality) d(x,z) is less than or equal to d(x,y) + d(y,z)
2) arbitrary intersections and finite unions of closed sets are closed
A ≠ ∅, B ≠ ∅, Ā ∩ B = ∅ and
B̄ ∩ A = ∅
Ā ∩ B = ∅ and B̄ ∩ A = ∅
E Is not connected if there exists a separation of E
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