Definition:Abstract Space

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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 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.