topological space (original) (raw)

A topological space is a set X together with a set 𝒯 whose elements are subsets of X, such that

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  ∅∈𝒯
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  X∈𝒯
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  If Uj∈𝒯 for all j∈J, then ⋃j∈JUj∈𝒯
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  If U∈𝒯 and V∈𝒯, then U∩V∈𝒯

Elements of 𝒯 are called open sets of X. The set 𝒯 is called a topology on X. A subset C⊂X is called a closed set if the complement X∖C is an open set.

A topology 𝒯′ is said to be finer (respectively, coarser) than 𝒯 if 𝒯′⊃𝒯 (respectively, 𝒯′⊂𝒯).

Examples

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  The discrete topology is the topology 𝒯=𝒫⁢(X) on X, where 𝒫⁢(X) denotes the power set of X. This is the largest, or finest, possible topology on X.
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  The indiscrete topology is the topology 𝒯={∅,X}. It is the smallest or coarsest possible topology on X.
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References

• 1 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
• 2 J. Munkres, Topology (2nd edition), Prentice Hall, 1999.