topological

Thân thiện

A mathematician draws a topological diagram on a chalkboard.

Definition

1. Adjective:
   • Relating to topology: Pertaining to the branch of mathematics concerned with the properties of geometric figures or spatial structures that remain unchanged under continuous deformations, such as stretching or bending, but not tearing or gluing.
   • Concerned with spatial relationships and connectivity: Involving the study of how objects are connected or arranged, independent of their specific shape or size.

Usage Examples

• Adjective:
  • The two shapes are considered equivalent in a topological sense because one can be deformed into the other without cutting.
  • A topological map focuses on the connections between locations rather than their precise geographical distances.
  • Understanding the topological properties of a network is crucial for analyzing its robustness.

Advanced Usage

• "Topological invariant": A property (like the number of holes) that remains constant for a geometric object under homeomorphisms (continuous deformations).
  • The genus of a surface is a key topological invariant.
• "Topological space": A fundamental concept in topology; a set of points along with a collection of neighborhoods for each point, satisfying specific axioms.
  • The definition of continuity is generalized for functions between topological spaces.

Variants and Related Words

• Topologically (adverb): In a manner relating to topology.
  • These two figures are topologically identical.
• Topology (noun): The mathematical field itself.
  • She is writing her thesis on algebraic topology.
• Topologist (noun): A mathematician specializing in topology.
  • The topologist explained the concept of a Möbius strip.

Synonyms

• Geometric (in a broad, qualitative sense, though not identical).
• Structural (relating to arrangement and form).

Related Phrases

• Topological defect: A discontinuity in a medium or field that cannot be removed by continuous transformation, often studied in physics.
  • Cosmic strings are a type of topological defect theorized in cosmology.
• Topological order: A phase of matter characterized by non-local properties and ground-state degeneracy, relevant in quantum computing.
  • Researchers are exploring materials that exhibit topological order for use in quantum bits.

Học thuật

Adjective

1. of or relating to topology

Từ đồng nghĩa

• topologic