# 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, *I*A: X → {0, 1}*I*A(x) = 1 if x ∈ A*I*A(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 *I*k≤ f ≤ *I*U 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.