Flashcards
§2 Interior, Closure, and Boundary
Cards in this deck (12)
- 1Definition 2.1: Interior (Int(A)) & Closure (Cl(A))Let X be a top. space & A⊆X.
The \(\underline {interior} \) of A
= the union of all open subsets of A
= ∪{B⊆A:B is open}
The \(\underline {closure} \) of A
= the intersection of all closed supersets of A.
= ∩{C: C⊇A & C is closed} - 2Theorem 2.4y ∈ Int(A)
iff there is an open set U with y∈U⊆A
iff y has a nbhd included in A - 3Theorem 2.5y ∈ Cl(A)
iff every open set containing y intersects A.
iff every nbhd of y contains a point of A. - 4Definition 2.3: DenseA subset A of space X is called \(\underline {dense} \) in X if Cl(A)=X.
- 5Properties of Interiors & Closures[On proof slides]
- 6Definition: ExteriorThe \(\underline {exterior} \) of A is Ext(A)=X-Cl(A)
Therefore Ext(A) = Int(X-A)
(the set of all points "not close" to A) - 7Definition 2.7: Limit Pointx is a \(\underline {limit-point} \) of A if every nbhd of x intersects A at a point other than x.
i.e. ∀ open U s.t. x∈U, ∃ y ∈ U∩A s.t. y≠x.
The set A'={x∈X: x is a limit point of A} is the \(\underline {derived-set} \) or \(\underline {derivative} \) of A. - 8Theorem 2.8Cl(A) = A∪A'
- 9Corollary 2.9A is closed iff A'⊆A
- 10Definition 2.13: BoundaryThe \(\underline {boundary} \) of A in space X is the set
δA = Cl(A) - Int(A)
= Cl(A)∩(X-Int(A)) - 11Theorem 2.14x∈δA iff every mbhd of x intersects both A & (X-A).
- 12Theorem 2.15: Properties of δ- δA closed
- δA=Cl(A)∩Cl(X-A)
- δA ∩ Int(A) = ∅
- δA ∪ Int(A) = Cl(A)
- A closed iff δA ⊆ A
- A open iff δA∩A = ∅
- A clopen iff δA = ∅
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- Diploma/Certificado de qualificação profissional de nivel 3 Mathematics (Topology (MATH313)) Flashcards on §2 Interior, Closure, and Boundary, created by Amy Wogan on 27/04/2016.
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