TopologySummary10f

# TopologySummary10f - INTRO to TOPOLOGY Setup 1 S G F K R...

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INTRO to TOPOLOGY Setup: 1 S,G,F,K R and ε > 0 and x,x 0 ,y R Neighborhood (NBHD) 2 N ε ( x 0 ) NTN = ε -NBHD of x 0 def = { y R : | x 0 - y | < ε } = ( x 0 - ε,x 0 + ε ) N 0 ε ( x 0 ) NTN = deleted ε -NBHD of x 0 def = { y R : 0 < | x 0 - y | < ε } = N ε ( x 0 ) \ { x 0 } = ( x 0 - ε,x 0 ) ( x 0 ,x 0 + ε ) S is a NBHD of x 0 def ⇐⇒ ε > 0 s.t. N ε ( x 0 ) S definitions and notation 3 x 0 is an interior point of S NTN ⇐⇒ x 0 S o def ⇐⇒ ∃ ε > 0 s.t. N ε ( x 0 ) S x 0 is a limit point of S NTN ⇐⇒ x 0 S 0 def ⇐⇒ ∀ ε > 0 : N 0 ε ( x 0 ) S 6 = x 0 is a boundary point of S NTN ⇐⇒ x 0 ∂S def ⇐⇒ ∀ ε > 0 : N ε ( x 0 ) S 6 = and N ε ( x 0 ) S C 6 = x 0 is an isolated point of S def ⇐⇒ [ x 0 S ] and [ ε > 0 s.t. N 0 ε ( x 0 ) S = ] ⇔ ∃ ε > 0 s.t. N ε ( x 0 ) S = { x 0 } x 0 is an exterior point to S def ⇐⇒ x 0 ` S C ´ o the boundary of S NTN = ∂S def = the set of all boundary points of S the closure of S NTN = S def

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TopologySummary10f - INTRO to TOPOLOGY Setup 1 S G F K R...

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