Completion requirements
Glossary of Mathematical Terms
A metric space is a set X equipped with a metric d: X × X →
R⁺ satisfying:
- d(x, y) = 0 iff x = y (identity of indiscernibles),
- d(x, y) = d(y, x) (symmetry),
- d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality).
Open Ball
In a metric space (X, d), the open ball centered at a ∈ X
with radius r > 0 is defined as:
B(a, r) = {x ∈ X : d(a, x) < r}.
Closed Ball
The closed ball centered at a with radius r is:
Bₓ(a, r) = {x ∈ X : d(a, x) ≤ r}.
Topological Space
A set X is a topological space if it is equipped with a
topology T, a collection of subsets of X (called open sets) that satisfies:
- X and ∅ are in T,
- The union of any collection of sets in T is also in T,
- The intersection of a finite number of sets in T is also in T.
Neighborhood
A subset N of a metric space (X, d) is called a neighborhood of a point x if there exists r > 0 such that B(x, r) ⊆ N.
Interior of a Set
The interior of a subset A ⊆ X, denoted Int(A), is the set of all points x ∈ A for which there exists r > 0 such that B(x, r) ⊆ A.
Closure of a Set
The closure of A, denoted Cl(A), is the set of all points x ∈ X such that every open ball B(x, r) intersects A.
Compact Space
A topological space is compact if every open cover of the space has a finite subcover.
Connected Space
A topological space is connected if it cannot be divided into two disjoint non-empty open subsets.
Hausdorff Space
A metric space (X, d) is Hausdorff if for any two distinct points x, y ∈ X, there exist disjoint neighborhoods U of x and V of y.
Boundary of a Set
The boundary of a subset A ⊆ X, denoted ∂A, is the set of all points x ∈ X such that every neighborhood of x intersects both A and X \ A.
Accumulation Point
A point x ∈ X is an accumulation point of A if every open ball centered at x contains at least one point of A different from x itself.
Isolated Point
A point x ∈ A is an isolated point if there exists r > 0 such that B(x, r) ∩ A = {x}.
Diameter of a Set
The diameter of a subset A of a metric space (X, d) is
defined as:
diam(A) = sup{d(x, y) : x, y ∈ A}.
Equivalent Metrics
Two metrics d₁ and d₂ on a set X are equivalent if they induce the same topology, i.e., they have the same collection of open sets.
Continuous Mapping
A function f: (X, dₓ) → (Y, dᵧ) between two metric spaces is
continuous at x₀ ∈ X if, for every ε > 0, there exists δ > 0 such that:
dₓ(x, x₀) < δ ⇒ dᵧ(f(x), f(x₀)) < ε.
Uniform Continuity
A function f: (X, dₓ) → (Y, dᵧ) is uniformly continuous if,
for every ε > 0, there exists δ > 0 such that for all x, y ∈ X:
dₓ(x, y) < δ ⇒ dᵧ(f(x), f) < ε.
Lipschitz Mapping
A function f: (X, dₓ) → (Y, dᵧ) is k-Lipschitz if there
exists a constant k > 0 such that:
dᵧ(f(x), f) ≤ k · dₓ(x, y) ∀x, y ∈ X.
Contraction Mapping
A function f: (X, dₓ) → (X, dₓ) is a contraction if there
exists 0 ≤ k < 1 such that:
dₓ(f(x), f) ≤ k · dₓ(x, y) ∀x, y ∈ X.
Isometry
A bijection f: (X, dₓ) → (Y, dᵧ) is an isometry if it
preserves distances, i.e.,
dᵧ(f(x), f) = dₓ(x, y) ∀x, y ∈ X.
Norm
A norm on a vector space X is a function ∥·∥: X → ℝ
satisfying:
- ∥x∥ ≥ 0 and ∥x∥ = 0 iff x = 0 (positivity),
- ∥αx∥ = |α| · ∥x∥ for all scalars α,
- ∥x + y∥ ≤ ∥x∥ + ∥y∥ (triangle inequality).
Path-Connected Space
A topological space X is path-connected if for any two points x, y ∈ X, there exists a continuous function f: [0, 1] → X such that f(0) = x and f(1) = y.