Definition

1. Noun:
   • (Mathematics) A set of points that satisfies a specific set of postulates or axioms: A "mathematical space" is a structured set, often with additional properties like a defined concept of distance or continuity, which serves as a framework for mathematical analysis and geometry.

Usage Examples

• Noun:
  • Euclidean space is a fundamental example of a mathematical space.
  • The properties of this mathematical space allow for the study of vector operations.
  • In topology, a mathematical space is defined by its open sets.

Advanced Usage

• "underlying mathematical space": Refers to the fundamental set structure upon which additional mathematical constructs are built.
  • The manifold's properties depend on the geometry of its underlying mathematical space.

Variants and Related Words

• Topological space (n): A type of mathematical space defined by a collection of open sets satisfying specific axioms.
  • A topological space is a more general concept than a metric space.
• Metric space (n): A mathematical space where a distance function (metric) is defined between any two points.
  • In a metric space, the distance between points must be zero only if the points are identical.
• Vector space (n): A mathematical space formed by vectors, where vector addition and scalar multiplication are defined.
  • Linear algebra primarily studies the properties of vector spaces.
• Space (n): In mathematics, often used as a shorter, informal term for a mathematical space when the context is clear.
  • The solution lies within a higher-dimensional space.

Synonyms

• Space (in a mathematical context).
• Structure (when referring to the axiomatic set with defined properties).

Related Phrases

• Phase space (n): A specific type of mathematical space in physics and dynamics used to represent all possible states of a system.
  • The system's evolution can be traced as a path in its phase space.
• State space (n): Similar to phase space, used in control theory and other fields to model all possible states.
  • The engineer analyzed the system's stability within its state space.

Notes on Meaning

The term "mathematical space" is a broad, overarching category. Its specific meaning is determined by the axioms (postulates) that define it. For instance, a "Euclidean space" has different defining axioms than a "Hilbert space." The reference to a "topological space" being "finite dimensional" is an example of a further property that can be ascribed to certain types of mathematical spaces.

Noun

1. (mathematics) any set of points that satisfy a set of postulates of some kind
   • assume that the topological space is finite dimensional