General Topology — Part 8 (Cheat Sheet)
Definition
Topological Space (X, τ):
T1: ø, X ∈ τ
T2: any (finite or infinite) union of sets in τ is itself in τ
T3: any finite intersection of sets in τ is itself in τ
CS1. ø, X are both closed sets
CS2. any finite union of closed sets is a closed set
CS3. any (finite or infinite) intersection of closed sets is a closed set
Bases B = {b| b ⊆ X}
Ba1. ∀x ∈ X, ∃b ∈ B such that x ∈ b ⊆ X,
Ba2. For b1, b2 ∈ B and x ∈ b1∩b2, ∃b3 ∈ B such that x ∈ b3 ⊆ b1∩b2.
τ = {o|o ⊆ X, ∀x ∈ o ∃b ∈ B s.t. x ∈ b ⊆ o}
Sub-bases S = {s| s ⊆ X} where ∪S = X
B = {b| b = ∩s[i], s[i] ∈ S}
Subspace Topology (Y, τY):
τY ={oY | oY= Y∩o, o ∈ τ, Y ⊆ X}
Bases BY = {Y∩b| b ∈ B, Y ⊆ X}
Sub-bases SY = {s’| s’=Y∩s, s ∈ S} where ∪SY = Y
BY = {b| b = ∩s’[i], s’[i] ∈ SY}
Neighbourhood N of point x:
∃o ∈ τ s.t. x ∈ o ⊆ N
Limit point x of set A:
∀ox ∈ τ ⇒ ox\{x}∩A ≠ø
LP1. A point x ∈ X is a Limit Point of a set A ⊆ X iff every Neighbourhood of x contains a point of A different from x.
Limit point x of sequence (z[k]) in X:
z[k] → x ⇔ ∀V∈N(x), ∃n∈N ∀k≥n z[k]∈V
Open Set A:
NB1. A is an Open Set iff A is a neighbourhood of all its element
NB2. Any finite intersection of neigbourhoods of x ∈ X is also a neigbourhood of x
OS1. A is a union of Bases elements.
OS2. A is a union of finite intersections of Sub-bases elements.
Denote the collection of all Limit Points of set A as A’.
Closure of set A, Cl(A):
CL1. Cl(A) = ∩{c|A ⊆ c ⊆ X, c is a closed set}
CL2. Cl(A) is the smallest Closed Set containing A
CL3. Cl(A) = {x|x∈X, ∀N ∈ N(x), N∩A ≠ø}
CL4. Cl(A) = A∪A’
Closed Set A:
CL5. A is closed in X iff Ac = X\A ∈ τ
CL6. A is closed in X iff A = Cl(A)
CL7. A is closed in X iff A contains all its Limit Point of set A: A’ ⊆ A
CS4. A is closed in Y iff A = Y∩W for some W closed in X where A ⊆ Y ⊆ X
CS5. A is closed in X iff A contains all the Limit Point of sequence (z[k]) ⊂ A
Dense Set A:
DS1. Cl(A) = X
DS2. X is the smallest Closed Set containing A
DS3. ∀x∈X ∀N ∈ N(x), N∩A ≠ø
DS4. A∪A’ = X
Normal Space (X, τ):
∀A,B⊆ X, Ac, Bc ∈ τ, A⋂B=ø ⇒∃o1, o2 ∈ τ s.t. A ⊆ o1 and B ⊆ o2 and o1⋂o2=ø
Dyadic rational number:
D(n) = {m/2^n | m, n ∈ N and 0 ≤m≤2^n-1 and n≥0}
Compact Space (X, τ):
∪Uα = X ⇒ ∪U[i] = X
Compact Set Y:
∪Uα ⊇ Y ⇒ ∪U[i] ⊇ Y
Hausdorff Space (X, τ):
∀p, q ∈ X, p ≠ q, ∃U, V ∈ τ s.t. p ∈ U, q ∈ V and U ∩ V = ∅
Compact Hausdorff Space (X, τ) is both Compact and Hausdorff
Locally Compact Space (X, τ):
∀p ∈ X, p has a compact neighbourhood. i.e. ∃U ∈ τ and a compact set K s.t. p ∈ U ⊆ K with U, K ⊆X
Locally Compact Hausdorff Space (X, τ) is Locally Compact and Hausdorff
One-point compactification (X*, τ*):
X* = X ∪ {∞}, τ* = τ ∪ { {∞} ∪ X\k | k is a closed compact set in X}
Support of Function
supp(f) = Cl({ x ∈ X| f(x)≠0 })
K(X) = { f ∈ C(x) | supp(f) is compact }
Properties
CT1. A function f : R → R is said to be continuous at x0 ∈ R if
∀ε>0, ∃δ>0 such that|x−x0|<δ ⇒|f(x)−f(x0)|< ε
CT2. A mapping f : X → Y is said to be Continuous if
∀ oY ∈ τY, f-1(oY) ∈ τX
CT3. A mapping f : X → Y is said to be Continuous at p ∈ X if
∀ Ny ∈ N(f(p)), ∃ Nx ∈ N(p) s.t. f(Nx) ⊆ Ny
CT4. Inverse image of every basis element of τY is open.
CT5. Inverse image of every subbasis element of τY is open.
CL8. A ⊆ B ⇒ Cl(A) ⊆ Cl(B)
CL9. A ⊆ Bc and B is open ⇒ Cl(A) ⊆ Bc
CL10. Cl(A∩B) ⊆ Cl(A) ∩ Cl(B)
CL11. Cl(A∪B) = Cl(A) ∪ Cl(B)
CL12. ClY(A) = Y ∩ ClX(A) where A ⊆ Y ⊆ X
CP1. A closed subset of a compact set is compact
CP2. Finite union of compact set is compact
CP3. The continuous mapping on the compact set is compact
CP4. A is compact in X iff A is compact in Y where A ⊆ Y ⊆ X
HD1. The compact set in the Hausdorff Space is closed
HD2. Rn is a Hausdorff Space
HD3. Let X be the Hausdorff Space. For any p ∈ X and compact set A ⊆ X and A does not contain p, there exists open set U and V where U ∩ V = ∅ and p ∈ U and A ⊆ V.
HD4. Let X be the Hausdorff Space. For any disjoint compact set A, B ⊆ X, there exists open set U and V where U ∩ V = ∅ and A ⊆ U and B ⊆ V.
HD5. Let X be the Hausdorff Space. For a compact set K ⊆ X, and open sets U1,U2 ⊆ X where K ⊆ U1 ∪ U2, there exists compact sets K1, K2 s.t. K=K1∪K2 where K1⊆U1, K2⊆U2
CH1. A Compact Hausdorff Space is a Normal Space.
CH2. A Compact [Hausdorff] Space is a Locally Compact [Hausdorff] Space.
Locally Compact Hausdorff Space (X, τ) can be extended to Compact Hausdorff Space (X*, τ*) by One-point compactification
Theorem
Lemma A:
For any closed set A ⊆ X and an open set C ∈ τ where A ⊆ C, there is an open set o∈τ s.t. A ⊆ o ⊆ Cl(o) ⊆ C
Urysohn’s lemma:
(X, τ) is a Normal Space iff for every two disjoint nonempty closed subsets A, B ⊆ X there is a continuous function f: X → [0,1] s.t. f(A)=0 and f(B)=1
f(x) = inf{p ∈ D⋂[0,1]| x ∈ o[p]} when x ∈ X\B
f(x) = 1 otherwise
Heine–Borel theorem:
HB. A is a compact subset in Rn iff A is closed and bounded
Extreme Value Theorem:
f is continuous on the closed interval [a,b]
⇒ ∃c,d ∈ [a,b] s.t. f(c)≥f(x)≥f(d)
K(X) denotes the set of f ∈ C(x) having Compact Support:
K(X) = { f ∈ C(x) | supp(f) is compact }
I denotes the indicator function:
Given a subset A ⊆ X, IA: X → {0, 1}
IA(x) = 1 if x ∈ A
IA(x) = 0 if x ∉ A
LCH1. Let X be a Locally Compact Hausdorff space. For any compact set k⊂X and let U be open set in X where k ⊆ U, there exists f ∈ K(X), satisfies Ik≤ f ≤ IU and k ⊆ supp(f) ⊆ U
LCH2. Let X be a Locally Compact Hausdorff space. Let f ∈ K(X) and let U1,U2, … Un be open subsets of X s.t. supp(f) ⊆ ∪{ Ui | 1≤i≤n }. Then, there are continuous f1,..,fn ∈ K(X) s.t. f = f1+..+fn and for each i, supp(fi) ⊆ Ui. Furthermore, if f is non-negative, then each fi can be chosen to be non-negative as well.
