Download Follow-up on derived sets

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Grothendieck topology wikipedia, lookup

General topology wikipedia, lookup

Transcript
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 โˆฉ ๐ต)โ—ฆ = (๐ดโ—ฆ ) โˆฉ (๐ต โ—ฆ ) for all sets ๐ด and ๐ต.
Union (๐ดโ—ฆ ) โˆช (๐ต โ—ฆ ) always is a subset of (๐ด โˆช 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 โˆฉ ๐ต) always is a subset of (Cl ๐ด) โˆฉ (Cl ๐ต) and sometimes is a proper subset.
Union Cl(๐ด โˆช ๐ต) = (Cl ๐ด) โˆช (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 โˆฉ ๐ต) and (Fr ๐ด) โˆฉ (Fr ๐ต) have no general relationship to each other.
Union Fr(๐ด โˆช ๐ต) always is a subset of (Fr ๐ด) โˆช (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) โˆฉ (Ext ๐ต) always is a subset of Ext(๐ด โˆฉ ๐ต) and sometimes is a proper
subset.
Union Ext(๐ด โˆช ๐ต) always is a subset of (Ext ๐ด) โˆช (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 โˆฉ ๐ต)โ€ฒ always is a subset of (๐ดโ€ฒ ) โˆฉ (๐ต โ€ฒ ) and sometimes is a proper subset.
Union (๐ด โˆช ๐ต)โ€ฒ = (๐ดโ€ฒ ) โˆช (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