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Definition 1.3. (v^k)k∈N converges to w ∈ R^n if
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Definition 1.3.1: If f : R^n → R and a ∈ R^n then we say that f is continuous at a
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define metric space
- a pair (X, d) of a set and a distance function d: X × X → R≥0 satisfying the axioms for a distance function. If the distance function is clear from context, we may, for convenience,
- simply write X rather than (X, d).
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Definition 2.4. Let (X, dX) and (Y, dY ) be metric spaces. A function f : X → Y
is said to be continuous at a ∈ X if
- for any e > 0 there is a δ > 0 such that for any x ∈ X with dX(a, x) < δ we have dY (f(x), f(a)) < e. We say f is continuous if it
- is continuous at every a ∈ X.
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If (xn)n≥1 is a sequence in X, and a ∈ X, then we say (xn)n≥1 converges to a
if,
for any e > 0 there is an N ∈ N such that for all n ≥ N we have dX(xn, a) < e.
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A function f : X → Y is said to be uniformly continuous if
- for any e > 0, there exists a δ > 0 such that for all x1, x2 ∈ X with dX(x1, x2) < δ
- we have dY (f(x1), f(x2)) < e.
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give three common metrics on R^n
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define an open ball and a closed ball
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define open, a neighbourhood and a topology
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define a topology, topological space and continuity of a function between two topological spaces
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define a limit point and an isolated point
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define the boundary of a subset of a metric space
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define homeomorphism, homeomorphic
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what is a Cauchy sequence
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define a complete metric space
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define Lipschitz map, contraction
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define disconnected, connected
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define the connected component
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define concatenation, opposite path
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define path-component of a metric space
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