distance:∣a−b∣
properties:(1)∣x∣≥0,for all x∈R,and "=”⇔x=0 (2):∣a−b∣=∣b−a∣(∣x∣=∣−x∣) (3):∣x+y∣≤∣x∣+∣y∣,for all x,y∈R
(∣a−c∣≤∣a−b∣+∣b−c∣)
度量空间
Distance function/metric space
Let X be a set. Def:A function X×X⟶dRis called a distance function on X 1.∀x,y∈X,d(x,y)≥0 and "=”⇔x=y 2.∀x,y∈X,d(x,y)=d(y,x) 3.∀x,y,z∈X,d(x,z)≤d(x,y)+d(y,z)
Example:
A:
1.x=(x1,x2,…,xm),y=(y1,y2,…,ym)∈Rn d2(x,y):=∣x1−y1∣2+⋯+∣xm−ym∣2=∣x−y∣ d2 is a metric on Rn(Cauchy inequality)
2.d1(x,y):=∣x1−y1∣+∣x2−y2∣+⋯+∣xm−ym∣
3.d∞(x,y)=max{∣x1−y1∣,…,∣xm−ym∣} B:
X:a set.For x,y∈X,let d(x,y):={10ififx≤yx=y d(x,y)⇒the discrete metric
开集，闭集
we may generalize the definitions about limits and convergence to metric space Def Let (X,d) be a metric space,an(n∈N)be a seq in X.and Lin X an(n∈N)converges to L
(1)For r≥0and x0∈X,we let Br(x0)={x∈X∣d(x,x0)≤r}(open ball)
(2).S is an open set(of(X,d)),if ∀x∈S,∃r>0
（Br(x0)⊂S）open ball ⇒open set
EX: (X,d):metric space.x0∈X,r≥0
Show that:(1)Br(x0)is open
(2){x∈X∣d(x,x0）>r}is open
warning:A subset S of a topological space (X,T) is said to be clopen if it is both open and closed in (X,T)
Example. Let X={a,b,c,d,e,f} and τ1={X,∅,{a},{c,d},{a,c,d},{b,c,d,e,f}}
We can see:
(i) the set {a} is both open and closed;
(ii) the set {b,c} is neither open nor closed;
(iii) the set {c,d} is open but not closed;
(iv) the set {a,b,e,f} is closed but not open.
In a discrete space every set is both open and closed, while in an indiscrete space(X,τ), all subsets of X except X and ∅ are neither open nor closed.
Topological space
Def:A topology space X=(X,ðx)consists of a set X,called the underlying space of X ,and a family ðxof subsets of X(ie.ðx⊂P(X)) P(X)means the power set of X
s.t.:(1):X and ∅∈ðx
(2):Uα∈ðx(α∈A)⇒ ∪α∈AUα∈ðx
(3).U,U′∈ðx⇒U∩U′∈ðx ðx is called a topology(topological structure) on X Convention:We usually use X to indicate the set Xand omit the subscript x in ðx by saying “a topological space(X,ð)” Examples:(1)metric space: (X,d)↬(X,ðd)(open sets induced by d) ∙Different distance funcs might determine the same topology Def:Let X and Y be topology spaces and X⟶fYa map.
We say that f is conti(from X to Y)
f is conti at a point x0∈X(from X to Y)
(1)if∀V∈f(x0),V∈ðY,∃x0∈U∈ðx,f(U)⊂V
(2)f is continuous(from X to Y)
if it is conti at every point of X Def: X : top space K⊂X
K is compact on X if ∀Uα⊂openX