Topological and Metrical Spaces Flashcards
A metric space (X,d) is a X together with a metric or distance function d : X × X → [0, ∞) satisfying …
d(x,y) = 0 if and only if x = y
d(x, y) = d(y, x)
d(x, z) ≤ d(x, y) + d(y, z)
A open ball is defined as …
B_r(x):={y∈X |d(x,y)0, x ∈ X
A norm p satisfies …
i) p(u + v) ≤ p(u) + p(v)
ii) p(av) = |a| p(v)
iii) If p(v) = 0 then v = 0
Based on metric spaces, a normed space is defined as …
a metric space (X, d) with d(x, y)=||x-y||
The discrete metric is …
(X,d) with X a set and d(x,y) = 1 - δ_x,y
with δ_x,y := if x=y then 1 else 0
the Kronecker-Delta
A topological space (X, T ) is a set X together
with a set T ⊂ 2X of subsets (called open sets)
satisfying …
i) ∅ ∈ T and X ∈ T
ii) union of Uα ∈ T for any family ({Uα} for α∈I) ⊂T.
iii) intersection of Uα ∈T for any family ({Uα} for α∈I) ⊂T with |I| < ∞
The cofinite topology on X is …
T :={U⊂X | U=∅ or |X\U| < ∞}
The trivial topology is …
T := {∅, X}
A family of open subsets is a basis of the topology if …
every open U ⊂ X is a union of subsets belonging to the family
A basis of the topology of a metric space (X,d) is …
the set of all open balls
Let (X, T) be a topological space.
The closure of M is …
cl(M) := intersection of all closed sets A containing M
Let (X, T) be a topological space.
The interior int(M) is …
int(M) := union of all open sets U being contained by M
Let (X, T) be a topological space.
A neighborhood N of an element x ∈ X is a subset N ⊂ X such that …
x∈U⊂N for an open set U∈T
A metric space is Hausdorff if
if for all x, y ∈ X such that x != y there exist open neighborhoods Ux and Uy of x and y, respectively, such that Ux ∩ Uy = ∅
