Science Metro

Metric-space

$function$

map

$Axiom of metric$

d : X \times X \to R

is a metric:

\forall x, y, z \in X

1. d (x, y) \geq 0 2. d (x, y) = 0 \Leftrightarrow x = y 3. d (x, y) = d (y, x) 4. d (x, z) \leq d (x, y) + d (y, z)

$Metric space$

Metric space :

(X, d)

$ε-neighborhood$

ε

-neighborhood around

x_{0}

U (x_{0}, ε) := {x \in X ∣ d (x, x_{0}) < ε}

$Neighborhood$

U

is neighborhood of

x

:

\exists ε > 0 (U (x, ε) \subset U)

$Open set$

A

is an open set

: \forall x \in 0\exists ε > 0 (U (x, ε) \subset O)

$Metric topology$

O_{d}

=

{O \subset X ∣\forall x \in A \exists ε > 0 (U (x, ε) \subset A)}

$Close set$

F

is close set :

F^{c} \in O_{d}

$Interior point$

x

is interior point of

A

:

\exists ε > 0 (U (x, ε) \subset A)

$Interior$

Interior of

A

A^{o} := {x \in X ∣ x

は

A

の内点

}

$Closure point$

x

is closure point of

A

:

\forall ε > 0 (U (x, ε) \cap A \neq = ϕ)

$Closure$

Closure of

A

\overline{A} := {x \in X ∣ x

is a closure point of

A}

$Exterior point$

x

is an exterior point of

A

:

x

is interior point of

\overline{A}

$Exterior$

Exterior of

A

A^{co} := {x \in X ∣ x

is exterior point of

A

} = (A^{c})^{o}

$Boundary$

Boundar of

A

:

\overline{A} \ A^{o}

$Accumulation point$

x

is an accumulation point of

A

:

x

が

A ∖ {x}

is closure point

$Isolated point$

x

is an isolated point of

A

:

\exists ε > 0 (U (x, ε) \cap A = {x})

$continuous function$

f

is continuous at

x

:

\forall ε > 0\exists δ > 0 (f (U (x, δ) \subset U (f (x, ε)))

f

is continuous:

\forall x \in X (f continuous at x)

$Corollary$

1. f is continuous at x 2. n \to \infty lim x_{n} = x \Rightarrow n \to \infty lim f (x_{n}) = f (x)