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:

\((\text M 1)\)	$:$			\(\ds \forall x \in A:\)	\(\ds \map d {x, x} = 0 \)
\((\text M 2)\)	$:$			\(\ds \forall x, y, z \in A:\)	\(\ds \map d {x, y} + \map d {y, z} \ge \map d {x, z} \)
\((\text M 3)\)	$:$			\(\ds \forall x, y \in A:\)	\(\ds \map d {x, y} = \map d {y, x} \)
\((\text M 4)\)	$:$			\(\ds \forall x, y \in A:\)	\(\ds x \ne y \implies \map d {x, y} > 0 \)

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): abstract space
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): abstract space