Apostle 3: Elements of Point Set Topology Flashcards
open n-ball of radius r and center a
B(a) or B(a;r)
open set
if and only if S = int S
interior point int S
a is called an interior point of S if there is an open n-ball with center at a, all of whose points belong to S
The status of union and intersection of open sets
- Union of any collection of open sets is an open set 2. The intersection of a finite collection of open sets is open
component interval
Let S be an open subset of R. An open interval I is called a component interval of S if I belong to S and there is no open interval J != I such that I belong to J belong to S
representation theorem for open sets on the real line
Every nonempty open set S in R is the union of a countable collection of disjoint component intervals of S
Bolzano-weierstrass
If a bounded set S in R contains infinitely many points, then there is at least one point in R which is an accumulation point of S
Lindelof covering theorem
A belongs to Rn, and F be an open covering of A. Then there is a countable subcollection of F which also covers A
covering
a collection F of sets is said to be a covering of a given set S if S is covered by the union of A, A belongs to F
open covering
If F is a collection of open sets and it covers S, then F is called an open covering of S
Heine-Borel theorem
Let F be an open covering of a closed and bound set A in Rn, then a finite subcollection of F also covers A
compact set
if, and only if, every open covering of S contains a finite subcover
Three equal statements
- S is compact 2. S is closed and bounded 3. Every infinite subset of S has an accumulation point in S
complete
A metric space (S,d) is called complete if every Cauchy sequence in S converges in S
