Metric space- Let X be a non-empty set and we have a real valued function ρ:X * X → ℝ such that, for all x,y,z ∈ X.
(a) ρ(x,y) ≥ 0
(b) ρ(x,y) = 0 iff x = y
(c) ρ(x,y) ≤ ρ(x,z)+ρ(z,y)
Then ρ is called metric and <X,ρ> is a metric space.
Pseudo metric- A metric ρ with the relaxation, ρ(x,y) = 0 for some x ≠ y.
Extended metric- A metric ρ such that
ρ:X * X → ℝ ∪ {∞}.
Subspace- <X, ρ> is a metric space, Y ⊂ X, then <Y, ρ> is a subspace of <X, ρ>.
Euclidean metric- x̅, y̅ ∈ X ⊂ℝn,
x̅ = (x1, ... , xn), y̅ = (y1, ... , yn),
ρ(x̅, y̅) = [Σ |xi - yi|]1/n, ∀ i = 1, 2, ... , n.
Convergence- A sequence {xn} converges to an element
x ∈ X if lim ρ(xn , x) = 0.
Open set- A set G ⊂ X is open if it contains a sphere about each of its points.
Closed set- A set F is closed if X ~ F is open.
Neighbourhood- A neighbourhood of a point x ∈ X is an open set which contains x0.
Interior point- x0 is an interior point of set A if A is a neighbourhood of x0.
Interior set- Interior of set A contains all interior points of A.
Limit point- If A ⊂ X and x0 ∈ X, then x0 is limit point of A, if every neighbourhood of x0 contains points of A distinct from x0.
Closure- Closure of set A is A̅ which contains all points which are either point of A or limit points of A.
Dense set- A, B ⊂ X, A is dense in B if A̅ ⊃ B.
Everywhere Dense- A is everywhere dense if A̅ = X.
Separable metric space- If metric space X has a countable subset which is everywhere dense, then it is separable metric space.
Open Covering- If C is a collection of open sets in metric space X with the property that every x ∈ X is a member of atleast one set G ∈C . TheC is called open covering of X.
Sub-covering- A sub-covering of the open coveringC is any sub-coveringC ' ⊂C which is also an open covering of X.
Cauchy sequence- A sequence {xn} in a metric space
X = (X, ρ) is Cauchy sequence, if for every ε > 0, there is N such that m, n > N
⇒ ρ(xm, xn) < ε.
Complete metric space- A metric space is complete, if every Cauchy sequence in metric space converges.
Contraction- If (X, ρ) is a metric space, a mapping
T : X → X is called contraction in X if there is a number K, with 0 < K < 1, such that x, y ∈ X, x ≠ y
⇒ ρ(Tx, Ty) ≤ Kρ(x, y).
Bolzano-Weierstrass property- A space X has Balzano-Weierstrass property if every infinite sequence in X has atleast one limit point.
Nowhere dense- If A ⊂ X is nowhere dense, A̅ the closure of A has no interior point.
First Category (Merger)- A set A ⊂ X is of first category in X, if it is the union of countably many nowhere dense sets in X.
Second category (Non-merger)- A set A ⊂ X is of second category in X, if it is not of first category.
Total boundedness- A metric space X is totally bounded if for every ε > 0, X contains a finite set, called an ε - net, such that the finite set of open spheres of radius ε and centres in the ε - net covers X.
Connected set- A ⊂ X, set A is connected if it cannot be represented as the union of two sets, each of which is disjoint from the closure of the other.
(a) ρ(x,y) ≥ 0
(b) ρ(x,y) = 0 iff x = y
(c) ρ(x,y) ≤ ρ(x,z)+ρ(z,y)
Then ρ is called metric and <X,ρ> is a metric space.
Pseudo metric- A metric ρ with the relaxation, ρ(x,y) = 0 for some x ≠ y.
Extended metric- A metric ρ such that
ρ:X * X → ℝ ∪ {∞}.
Subspace- <X, ρ> is a metric space, Y ⊂ X, then <Y, ρ> is a subspace of <X, ρ>.
Euclidean metric- x̅, y̅ ∈ X ⊂ℝn,
x̅ = (x1, ... , xn), y̅ = (y1, ... , yn),
ρ(x̅, y̅) = [Σ |xi - yi|]1/n, ∀ i = 1, 2, ... , n.
Convergence- A sequence {xn} converges to an element
x ∈ X if lim ρ(xn , x) = 0.
Open set- A set G ⊂ X is open if it contains a sphere about each of its points.
Closed set- A set F is closed if X ~ F is open.
Neighbourhood- A neighbourhood of a point x ∈ X is an open set which contains x0.
Interior point- x0 is an interior point of set A if A is a neighbourhood of x0.
Interior set- Interior of set A contains all interior points of A.
Limit point- If A ⊂ X and x0 ∈ X, then x0 is limit point of A, if every neighbourhood of x0 contains points of A distinct from x0.
Closure- Closure of set A is A̅ which contains all points which are either point of A or limit points of A.
Dense set- A, B ⊂ X, A is dense in B if A̅ ⊃ B.
Everywhere Dense- A is everywhere dense if A̅ = X.
Separable metric space- If metric space X has a countable subset which is everywhere dense, then it is separable metric space.
Open Covering- If C is a collection of open sets in metric space X with the property that every x ∈ X is a member of atleast one set G ∈C . TheC is called open covering of X.
Sub-covering- A sub-covering of the open coveringC is any sub-coveringC ' ⊂C which is also an open covering of X.
Cauchy sequence- A sequence {xn} in a metric space
X = (X, ρ) is Cauchy sequence, if for every ε > 0, there is N such that m, n > N
⇒ ρ(xm, xn) < ε.
Complete metric space- A metric space is complete, if every Cauchy sequence in metric space converges.
Contraction- If (X, ρ) is a metric space, a mapping
T : X → X is called contraction in X if there is a number K, with 0 < K < 1, such that x, y ∈ X, x ≠ y
⇒ ρ(Tx, Ty) ≤ Kρ(x, y).
Bolzano-Weierstrass property- A space X has Balzano-Weierstrass property if every infinite sequence in X has atleast one limit point.
Nowhere dense- If A ⊂ X is nowhere dense, A̅ the closure of A has no interior point.
First Category (Merger)- A set A ⊂ X is of first category in X, if it is the union of countably many nowhere dense sets in X.
Second category (Non-merger)- A set A ⊂ X is of second category in X, if it is not of first category.
Total boundedness- A metric space X is totally bounded if for every ε > 0, X contains a finite set, called an ε - net, such that the finite set of open spheres of radius ε and centres in the ε - net covers X.
Connected set- A ⊂ X, set A is connected if it cannot be represented as the union of two sets, each of which is disjoint from the closure of the other.
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