The real part (one side) must equal the real part of the other side, and the coefficients of the imaginary parts of both sides must be equal. this gives us
a + c + (b + d)i = 0 + 0i -> a + c = 0 and b + d = 0 -> c=-a d=-b
the additive inverse of z was defined as as c+di. –> F = -a +(-b)i -> F=-a-bi
Closed union of subsets in a compact space
Proposition: A set is connected if and only if it is an interval, that is, of the form [a,b], [a,b), or (a,b) where a or b can be plus or minus infinity.
Proposition: If A in n-dimensional R space is open and connected, then it is path-connected.
compactness-containing-union-of-subsets-lemma-bolzano-weierstrass-theorem-subsequence
compactness in terms of a collection of sets and subsets. converges to a point in the domain of the metric space.
Heine Borel Theorem: a set in R is compact if and only if it is closed and bounded. Different concepts of compactness can be equivalent in a metric space. One such concept uses the idea of an open cover. Let M be a metric space and A⊆M
i) A cover of A is a family V of sets such that A ⊆ U V. V is an open cover if all sets in V are open. a subcover of the cover V is a subcollection of V whose union also contains A. subcover is finite if it contains only finitely many sets.
I’m a graduate of Dalhousie’s hon. math department. I’m glad to help out !