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Definition 1.3. (v^k)k∈N converges to w ∈ R^n if

Definition 1.3.1: If f : R^n → R and a ∈ R^n then we say that f is continuous at a


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).

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.

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.

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.

give three common metrics on R^n


what is

define an open ball and a closed ball

define open, a neighbourhood and a topology


define a topology, topological space and continuity of a function between two topological spaces


define a limit point and an isolated point


define the boundary of a subset of a metric space


define homeomorphism, homeomorphic

what is a Cauchy sequence

define a complete metric space


define Lipschitz map, contraction


define disconnected, connected

define the connected component



define concatenation, opposite path

define pathcomponent of a metric space

