Topology

Munkres, J. R. (2000). Topology. Prentice Hall.

Type

Book chapter

Intellectual base

Textbook

Year

2000

ISBN

9780131816299

Link

https://www.pearson.com/en-us/subject-catalog/p/topology/P200000006097

Cited from

mathematics

Cited by

20 citations across 20 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Boundedness
  • Boundedness is not the same as closure, as Munkres (2000) makes clear in his treatment of separation and finiteness properties in topological spaces.

    This sourceauthoritative topology text treating closure, separation, and the metric/topology relationship — substantiates that boundedness (a magnitude property) and closure are distinct structural properties.

• Closure
  • A team is closed under decision authority with respect to a designated class of decisions when the team can resolve any decision in that class without external escalation; jurisdictional closure is the analogous property for legal forums (a court is closed with respect to a class of cases when it can resolve any such case within its own procedures); and the design of organisational charters typically specifies the closure profile of each unit (which decisions are within the unit's authority, which require escalation), a discipline Galbraith (1973) develops in his canonical treatment of designing decision-authority structures.

    This sourceStandard topology textbook developing the closure operator, interior, boundary, and derived-set operators and the characterisation of continuity via closure-preservation.

• Connectedness
  • Topological connectedness and continuity interact (continuous images of connected sets stay connected) but answer different questions.

    This sourceSection 23 defines topological connectedness (no separation into two disjoint nonempty open sets) and proves continuous images of connected sets are connected.

• Continuity
  • and by the preimage-of-open-is-open condition in general topology, as systematically presented by Munkres (2000)

    This sourceStandard graduate-level topology textbook; develops the preimage-of-open-is-open definition of continuity for general topological spaces, the construction of homeomorphisms, and the framework relating metric, topology, and uniform structure as alternative notions of closeness.

• Equivalence Relation
  • An equivalence relation is not the same as a similarity measure or a thresholded similarity, a distinction Munkres (2000) makes operative in the metric-topology setting where ε-balls are reflexive and symmetric but transitivity fails for thresholded proximity.

    This sourceDevelops metric topology, open ε-balls, and continuity; ε-ball proximity is reflexive and symmetric but thresholded proximity is not transitive.

• Local-to-Global Aggregation
  • Topology (compactness) — any open cover admits a finite subcover; local properties on a compact space pass to the whole.

    This sourceCompactness as every open cover admitting a finite subcover; the topological sibling of logical compactness.

• Neighborhood
  • In mathematics the neighborhood of a point is the primitive of topology itself: open sets, continuity, limits, and convergence are all defined locally, and advanced structures (manifolds, sheaves) are neighborhood-glued.

    This sourceStandard reference defining the neighborhood as the primitive of point-set topology — open sets, continuity, limits, and convergence all defined locally — and the local-to-global construction of manifolds.

• Preimage
  • A preimage is only as trustworthy as the forward map is complete. Not an inverse function. An inverse exists only when every preimage is a singleton (the mapping is injective).

    This sourceDefines the preimage f^{-1}(U), proves it distributes over union/intersection/complement, and characterizes continuity as the preimage of every open set being open.

• Topology
  • a topological space is a carrier set \(X\) together with a designated collection \(\tau \subseteq \mathcal{P}(X)\) of open sets satisfying three axioms — both \(X\) and \(\emptyset\) are in \(\tau\); arbitrary unions of members of \(\tau\) are in \(\tau\); finite intersections of members of \(\tau\) are in \(\tau\) — and the entire substantive theory of continuity, convergence, connectedness, compactness, and homotopy is built from those axioms by definitions that refer only to the open-set system and never to a metric.

    This source(Standard graduate-level topology textbook; develops the preimage-of-open-is-open definition of continuity for general topological spaces, the construction of homeomorphisms, and the framework relating metric, topology, and uniform structure as alternative notions of closeness.)

Domain-specific

• Cover (topology)
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Gδ Set
  • In real analysis, the set of continuity points of a function from a topological space to a metric space is Gδ
• Graph of a Function
  • Standard topology texts treat products and graphs as subset constructions supporting such closure questions.
• Hausdorff Space
  • This is the \(T_2\) separation condition.
• Index set
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• K-Topology
  • Equivalently, start with the usual topology on \(\mathbb R\) and declare \(\mathbb R\setminus K\) open; intersections with usual intervals then produce the new basic neighborhoods.
• Metrizable space
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Number line
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Open and closed maps
  • An open map sends open sets to open sets; a closed map sends closed sets to closed sets.
• Regular space
  • Some authors include the T1 or Hausdorff condition in regular, while others reserve T3 for regular plus Hausdorff; the convention must be stated.
• Sequentially compact space
  • A space is sequentially compact when every sequence of its points contains a subsequence converging to a point in the space.

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Links previously used in the corpus

Before the registry existed this work was also linked 3 other ways.

• https://www.pearson.com/en-us/subject-catalog/p/topology-classic-version/P200000006299 ×1
• https://www.pearson.com/en-us/subject-catalog/p/topology/P200000003209 ×1
• https://www.pearson.com/en-us/subject-catalog/p/topology/P200000006299 ×1

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