Topology

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

Type

Book

Intellectual base

Textbook

Year

2000

ISBN

9780131816299

Link

https://openlibrary.org/isbn/9780131816299

Cited from

mathematics

Cited by

35 citations across 34 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.

• Indexed family
  • This conserved decision sequence is the basis for Prime status.
• 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

• Adherent point
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Ambient space (mathematics)
  • … observable transfers from A circle considered in the plane separates an inside from an outside, while the same abstract circle considered alone has no ambient complement. to A geometric proof states whether isotopy, distance or orientation is intrinsic to the subspace or inherited from its ambient manifold..
• Closed Set
  • Closedness is also preserved under arbitrary intersections and finite unions

    This sourceStates that arbitrary intersections of closed sets are closed and that finite unions of closed sets are closed.

    SupportedVerified against a saved copy of the source

    “(2) Arbitrary intersections of {{{closed}}} sets are {{{closed}}}”

    Read by an automated reader; not a human review. How support was checked

• Cover (topology)
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Cylinder set
  • … between constitutive identity and a convenient observable transfers from In an infinite coin-toss space, the event that tosses one and three are heads is a cylinder set regardless of all other tosses. to A probability construction distinguishes the cylinder algebra from its completed or generated sigma-algebra..
• Excisive triad
  • … cover every point of X. The scope is broad within that domain but bounded by the need for A and B are subspaces of X and their X-relative interiors cover every point of X. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• First-countable space
  • 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

    SupportedVerified against the source

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• 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.
• Image (of a Function)
  • In topology, the image of a continuous map inherits topological properties under appropriate hypotheses — a continuous map sends compact sets to compact sets, connected sets to connected sets

    This sourceContinuous images of connected spaces are connected and continuous images of compact spaces are compact.

    SupportedVerified against the source

    Read by an automated reader; not a human review. How support was checked

• Isolated point
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Locally simply connected space
  • … The circle is locally simply connected because sufficiently short open arcs are contractible, even though a loop around the entire circle is not null-homotopic. to Topological manifolds and CW complexes have locally contractible neighborhoods under their standard hypotheses and hence are locally simply connected..
• 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.
• Open Set
  • The structural payoff is that the same body of theorems — separation axioms, the Urysohn lemma, Tychonoff's theorem — applies uniformly across all these settings, with no re-derivation per context.

    This sourceMunkres's general-topology text, in which the Urysohn lemma and the Tychonoff theorem are proved once for topological spaces rather than re-derived for each metric setting. Munkres's contrast of the discrete and standard topologies on the reals as the illustration that openness is relative to the chosen collection τ. Munkres's statement of the axioms on τ - empty set and whole space open, arbitrary unions and finite intersections of open sets open.

    Supported in partVerified against the work's full text

    “Ascoli’s theorem. §37 The [Tychonoff] Theorem Like the [Urysohn] lemma, the [Tychonoff] theorem is what we call”

    Read by an automated reader; not a human review. How support was checked

• 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.
  • … set and exterior point, two open neighborhoods, containment and disjointness, equivalent neighborhood-closure condition, T1 or Hausdorff qualification and examples and counterexamples are explicit. to An applied instance preserves the invariant under changed notation, scale, dataset, jurisdiction, or implementation..
• Sequentially compact space
  • A space is sequentially compact when every sequence of its points contains a subsequence converging to a point in the space.
• Topological Space
  • The constructive interventions are the standard space-builders — form the subspace topology on a subset, the product topology on a Cartesian product, the quotient topology by an equivalence relation — each predicted to yield a new topological space whose continuity, compactness, and connectedness are derivable from the construction (Tychonoff guarantees a product of compact spaces is compact; a quotient of a connected space is connected)

    This sourceThe subspace, product and quotient topologies, Tychonoff's theorem that a product of compact spaces is compact, and the connectedness of continuous images such as quotients.

    Supported in partVerified against the source

    Read by an automated reader; not a human review. How support was checked

• Universal property
  • Category theory and algebra — the home turf: products, coproducts, limits, colimits, free objects, and adjoint functors are all defined by universal properties and determined up to unique isomorphism. Topology — the one-point and Stone–Čech compactifications and the universal cover are each defined by a universal mapping property among compactifications or covers

    This sourceThe one-point compactification (§29), the Stone-Cech compactification (§38) and the universal covering space (§80).

    Supported in partVerified against the source

    Read by an automated reader; not a human review. How support was checked

Mechanisms

• Gluing or Recomposition Workflow
  • The formal license for gluing is the pasting lemma.

    This sourceStates the pasting lemma that compatible continuous maps on overlapping pieces glue into a continuous map on their union.

• Patch Cover Inventory
  • The mathematical shadow of this object is an open cover.

    This sourceDefines an open cover as a collection of open subsets whose union contains the whole space.

• Quotient-Space Construction
  • Its failure mode is false collapse — quotienting by a relation that folds together cases which are actually different, so a material distinction vanishes into a class and can never be recovered.

    This sourceDefines a quotient space by mapping points in the same partition element to one equivalence class, thereby identifying equivalent points.

Verification

Does it exist? Confirmed. This work's ISBN resolves to a registered record, which fixes its identity. That is all it fixes.

Does it back the claim? Read against the text for 6 of 35 citations: 3 supported, 3 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.

Was it audited? Yes. A second, independent pass read the citation against the article text and recorded a verdict.

Support is checked per citation rather than per work — the same source can be cited soundly in one article and wrongly in another. Per-citation recording began recently, so a citation with no recorded check is a gap in the record rather than evidence it went unchecked.

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Links previously used in the corpus

Before the registry existed this work was also linked 4 other ways.

• https://obnb.uk/p11296815-topology ×2
• https://search.worldcat.org/title/Topology/oclc/42683260 ×1
• https://www.pearson.com/en-us/subject-catalog/p/topology-classic-version/P200000006299 ×1
• https://www.pearson.com/en-us/subject-catalog/p/topology/P200000006299 ×1

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