Math 2.0

General topology Definition

Subspace topology

Builds on Topological spaceIntersectionSubset

Definition

Let (X,τ)(X, \tau) be a topological space and let S⊆XS \subseteq X. The subspace topology on SS is τS={S∩U:U∈τ}.\tau_S = \{ S \cap U : U \in \tau \}. With this topology, SS is a subspace of XX.

Remarks

The set τS\tau_S is a topology on SS, because intersection with SS commutes with unions and finite intersections. A set open in SS need not be open in XX. For example, [0,1)[0, 1) is open in the subspace [0,1][0, 1] of ℝ\mathbb{R}, because [0,1)=[0,1]∩(−1,1)[0, 1) = [0, 1] \cap (-1, 1). For a metric space, the subspace topology is the metric topology of the subspace metric. The subspace topology is the coarsest topology on SS for which the inclusion map is continuous.

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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 spacedefinitionSubspace topology, definition, no evidence levelSubspace topologydefinition
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  17. Subspace topology definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Subspace_topology, Definition
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