# Definition:Abstract Space

## Definition

An abstract space is:

a set of objects

together with:

a set of axioms which define operations on and relations between those objects.

### Metric Space

A metric space $M = \struct {A, d}$ is an ordered pair consisting of:

$(1): \quad$ a non-empty set $A$

together with:

$(2): \quad$ a real-valued function $d: A \times A \to \R$ which acts on $A$, satisfying the metric space axioms:
 $(\text M 1)$ $:$ $\displaystyle \forall x \in A:$ $\displaystyle \map d {x, x} = 0$ $(\text M 2)$ $:$ $\displaystyle \forall x, y, z \in A:$ $\displaystyle \map d {x, y} + \map d {y, z} \ge \map d {x, z}$ $(\text M 3)$ $:$ $\displaystyle \forall x, y \in A:$ $\displaystyle \map d {x, y} = \map d {y, x}$ $(\text M 4)$ $:$ $\displaystyle \forall x, y \in A:$ $\displaystyle x \ne y \implies \map d {x, y} > 0$

### Topological Space

Let $S$ be a set.

Let $\tau$ be a topology on $S$.

That is, let $\tau \subseteq \powerset S$ satisfy the open set axioms:

 $(\text O 1)$ $:$ The union of an arbitrary subset of $\tau$ is an element of $\tau$. $(\text O 2)$ $:$ The intersection of any two elements of $\tau$ is an element of $\tau$. $(\text O 3)$ $:$ $S$ is an element of $\tau$.

Then the ordered pair $\struct {S, \tau}$ is called a topological space.

The elements of $\tau$ are called open sets of $\struct {S, \tau}$.

### Vector Space

Let $\struct {K, +_K, \times_K}$ be a division ring.

Let $\struct {G, +_G}$ be an abelian group.

Let $\struct {G, +_G, \circ}_K$ be a unitary $K$-module.

Then $\struct {G, +_G, \circ}_K$ is a vector space over $K$ or a $K$-vector space.

That is, a vector space is a unitary module whose scalar ring is a division ring.