Point-Set Topology

A Point-Set Topology is a Topology that deals with constructions and definitions of the Set Theory.

• AKA: General Topology.
• Example(s):
  • Order Topology,
  • Alexandrov Topology,
  • Co-kappa Topology,
  • Cocountable topology,
  • Cofinite Topology,
  • Discrete topology,
  • Double-Pointed Cofinite Topology,
  • Zariski topology,
• Counter-Example(s):
  • Algebraic Topology,
  • Differential Topology,
  • Geometric Topology.
• See: Metric (Mathematics), Topology, Set Theory, Differential Topology, Geometric Topology, Algebraic Topology, Continuous Function, Compact Set, Connected Set, Open Set, Topological Space, Metric Space.

---

---

References

2019a

• (Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/General_topology Retrieved:2019-4-25.
  • In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.

    The fundamental concepts in point-set topology are continuity, compactness, and connectedness:

    • Continuous functions, intuitively, take nearby points to nearby points.
    • Compact sets are those that can be covered by finitely many sets of arbitrarily small size.
    • Connected sets are sets that cannot be divided into two pieces that are far apart.
  • The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of open sets. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.

    Metric spaces are an important class of topological spaces where a real, non-negative distance, also called a metric, can be defined on pairs of points in the set. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.

2019b

• (Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/Topology#General_topology Retrieved:2019-4-25.
  • General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.

    The basic object of study is topological spaces, which are sets equipped with a topology, that is, a family of subsets, called open sets, which is closed under finite intersections and (finite or infinite) unions. The fundamental concepts of topology, such as continuity, compactness, and connectedness, can be defined in terms of open sets. Intuitively, continuous functions take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words nearby, arbitrarily small, and far apart can all be made precise by using open sets. Several topologies can be defined on a given space. Changing a topology consists of changing the collection of open sets, and this changes which functions are continuous, and which subsets are compact or connected.

     Metric spaces are an important class of topological spaces where distances between any two points are defined by a function called a metric. In a metric space, an open set is a union of open disks, where an open disk of radius r centered at x is the set of the points whose distance to x is less than d. Many common spaces are topological space whose topology can be defined by a metric. This is the case of the real line, the complex plane, real and complex vector spaces and Euclidean spaces. Having a metric simplifies many proofs.

2019c

• (Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/List_of_examples_in_general_topology Retrieved:2019-4-25.
  • This is a list of useful examples in general topology, a field of mathematics.
    • Alexandrov topology.
    • Cantor space.
    • Co-kappa topology
      • Cocountable topology.
      • Cofinite topology.
    • Compact-open topology.
    • Compactification.
    • Discrete topology.
    • Double-pointed cofinite topology.
    • Extended real number line.
    • Finite topological space.
    • Hawaiian earring.
    • Hilbert cube.
    • Irrational cable on a torus.
    • Lakes of Wada.
    • Long line.
    • Order topology.
      • Lexicographical/dictionary order.
      • Ordinal number topology.
      • Real line.
      • Split interval.
    • Overlapping interval topology.
    • Moore plane.
    • Sierpiński space.
    • Sorgenfrey line.
    • Sorgenfrey plane.
    • Space-filling curve.
    • Topologist's sine curve.
    • Trivial topology.
    • Unit interval.
    • Zariski topology.

---