Topological Space

A Topological Space is a mathematical space equipped with a mathematical structure defined by open sets satisfying the following three topological axioms: closure under arbitrary unions and finite intersections, and inclusion of the whole space and empty set.

• Context:
  • It can (often) define continuous functions between topological spaces.
  • ...
  • It can range from being a Metrizable space, where a metric can define the topology, to more Abstract Topological Spaces like Zariski topology used in algebraic geometry.
  • It can range from being a Compact Topological Space to being a Non-Compact Topological Space.
  • ...
  • It can facilitate the use of topological methods to analyze continuity, convergence, and connectedness.
  • It can serve as the foundation for various studies in mathematics, including algebraic topology, analysis, and differential geometry.
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• Example(s):
  • Basic Topological Spaces:
    • Euclidean Spaces with the standard topology.
    • Discrete Topological Spaces, where every subset is open.
    • Indiscrete Topological Spaces, where only the entire set and the empty set are open.
    • Subspace Topological Spaces, which is induced on subsets of a given topological space.
  • Specialized Topological Spaces:
    • Zariski Topology
    • Metric Spaces:
      • [math]\displaystyle{ \mathbb{R}^n }[/math] with the standard Euclidean metric
      • The space of continuous functions with the uniform metric
    • Measure Spaces:
      • The real line [math]\displaystyle{ \mathbb{R} }[/math] with the Borel sigma-algebra
      • The unit interval [math]\displaystyle{ [0, 1] }[/math] with the Lebesgue measure
    • Manifold Spaces:
      • Topological Manifolds:
        • The Möbius strip
        • The Klein bottle
        • Arbitrary-dimensional spheres S^n
        • Real projective spaces RP^n
        • Topological groups (e.g., the circle group S^1)
  • ...
• Counter-Example(s):
  • Sets with open sets that do not satisfy the topology axioms, such as unions or intersections that do not result in an open set.
  • Metric Spaces considered without their open sets; while every metric space can be a topological space, not every concept in metric spaces directly translates to topological spaces without context.
• See: Continuous Function, Compact Space, Metrizable Space, General Topology, Mathematics, Geometry, Closeness (Mathematics), Distance (Mathematics), Set (Mathematics), Point (Geometry), Topology, Neighbourhood (Mathematics), Open Set, Space (Mathematics).

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References

2024

• (Wikipedia, 2024) ⇒ https://en.wikipedia.org/wiki/topological_space Retrieved:2024-9-2.
  • In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate.

    A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds.

    Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spaces in their own right is called point-set topology or general topology.

2009

• http://en.wikipedia.org/wiki/Topological_space
  • Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. They appear in virtually every branch of modern Mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called Topology.

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