# Definition:Abstract Space

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## Definition

An **abstract space** is:

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:

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 field.

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 field.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**abstract space** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**abstract space**