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```Math 436
Introduction to Topology
Spring 2014
Follow-up on derived sets
Class on February 4 studied how intersection, union, and complement interact with interior,
closure, frontier, exterior, and the derived set in a topological space ๐. Here is a summary.
Interior
Intersection (๐ด โฉ ๐ต)โฆ = (๐ดโฆ ) โฉ (๐ต โฆ ) for all sets ๐ด and ๐ต.
Union (๐ดโฆ ) โช (๐ต โฆ ) always is a subset of (๐ด โช ๐ต)โฆ and sometimes is a proper subset.
Complement (๐ โงต ๐ด)โฆ always is a subset of ๐ โงต (๐ดโฆ ) and sometimes is a proper subset.
Closure
Intersection Cl(๐ด โฉ ๐ต) always is a subset of (Cl ๐ด) โฉ (Cl ๐ต) and sometimes is a proper subset.
Union Cl(๐ด โช ๐ต) = (Cl ๐ด) โช (Cl ๐ต) for all sets ๐ด and ๐ต.
Complement ๐ โงต (Cl ๐ด) always is a subset of Cl(๐ โงต ๐ด) and sometimes is a proper subset.
Frontier
Intersection Fr(๐ด โฉ ๐ต) and (Fr ๐ด) โฉ (Fr ๐ต) have no general relationship to each other.
Union Fr(๐ด โช ๐ต) always is a subset of (Fr ๐ด) โช (Fr ๐ต) and sometimes is a proper subset.
Complement Fr(๐ โงต๐ด) and ๐ โงต(Fr ๐ด) are complements of each other, since Fr(๐ โงต๐ด) = Fr ๐ด.
Exterior
Intersection (Ext ๐ด) โฉ (Ext ๐ต) always is a subset of Ext(๐ด โฉ ๐ต) and sometimes is a proper
subset.
Union Ext(๐ด โช ๐ต) always is a subset of (Ext ๐ด) โช (Ext ๐ต) and sometimes is a proper subset.
Complement Ext(๐ โงต ๐ด) always is a subset of ๐ โงต (Ext ๐ด) and sometimes is a proper subset.
Derived set
Intersection (๐ด โฉ ๐ต)โฒ always is a subset of (๐ดโฒ ) โฉ (๐ต โฒ ) and sometimes is a proper subset.
Union (๐ด โช ๐ต)โฒ = (๐ดโฒ ) โช (๐ต โฒ ) for all sets ๐ด and ๐ต.
Complement (๐ โงต ๐ด)โฒ and ๐ โงต (๐ดโฒ ) have no general relationship to each other.
February 6, 2014
Page 1 of 1
Dr. Boas
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