Math 2.0

General topology Definition

Comparison of topologies

Builds on Topological spaceSubset

Definition

Let XX be a set and let τ1\tau_1 and τ2\tau_2 be topologies on XX. If τ1⊆τ2\tau_1 \subseteq \tau_2, then τ1\tau_1 is coarser (or weaker) than τ2\tau_2, and τ2\tau_2 is finer (or stronger) than τ1\tau_1. If also τ1≠τ2\tau_1 \neq \tau_2, then τ1\tau_1 is strictly coarser and τ2\tau_2 is strictly finer.

Remarks

On each set XX, the indiscrete topology is the coarsest topology and the discrete topology is the finest topology. We have τ1⊆τ2\tau_1 \subseteq \tau_2 if and only if the identity map (X,τ2)→(X,τ1)(X, \tau_2) \to (X, \tau_1) is continuous. Two topologies need not be comparable: on {a,b}\{a, b\}, the topologies {∅,{a},X}\{\varnothing, \{a\}, X\} and {∅,{b},X}\{\varnothing, \{b\}, X\} are not comparable.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionTopological space, definition, no evidence levelTopological spacedefinitionComparison of topologies, definition, no evidence levelComparison of topolog…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Conjunction definition
  5. Disjunction definition
  6. Implication definition
  7. Predicate definition
  8. Empty set object
  9. Family of sets definition
  10. Intersection definition
  11. Union definition
  12. Existential quantifier definition
  13. Universal quantifier definition
  14. Subset definition
  15. Union of a family of sets definition
  16. Topological space definition
  17. Comparison of topologies definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Comparison_of_topologies, Definition
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DefinesComparison of topologies
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