Exercise · ZW-BOUNDARY-OF-A-SET-007 · ★★☆ Intermediate
In a topological space \(X\), which statement holds for every \(A\subseteq X\)?
Choose one option.
Show answer
- B. \(\partial A=\partial(X\setminus A)\)
Explanation
Since \(\partial A=\overline{A}\cap\overline{X\setminus A}\), the expression is symmetric in \(A\) and its complement. The others fail: for instance, \(\partial A\) never meets \(\operatorname{int}A\).
Review the concept: Boundary of a set
The boundary \(\partial A\) of a subset \(A\) of a topological space \(X\) is the set of points of the closure of \(A\) that do not belong to its interior.
\[\partial A=\overline{A}\setminus\operatorname{int}A\]
More practice problems: boundary of a set
- ZW-BOUNDARY-OF-A-SET-001 In X=\1,2,3\ with T=\∅,\1\,\1,2\,\1,2,3\\, what is the boundary of \1,3\?
- ZW-BOUNDARY-OF-A-SET-002 In X=\a,b,c,d\ with T=\∅,\a\,\b\,\a,b\,\a,b,c\,\a,b,c,d\\, what is the boundary of \a,c\?
- ZW-BOUNDARY-OF-A-SET-003 In X=\1,2,3,4\ with T=\∅,\1\,\3\,\1,3\,\1,2,3\,\1,3,4\,\1,2,3,4\\, what is the boundary of \1,2\?
- ZW-BOUNDARY-OF-A-SET-004 In ℝ with the usual topology, what is the boundary of [0,1]?
- ZW-BOUNDARY-OF-A-SET-005 In ℝ with the usual topology, what is the boundary of ℚ?
- ZW-BOUNDARY-OF-A-SET-006 In X=\1,2,3,4\ with T=\∅,\1,2\,\3,4\,\1,2,3,4\\, which of these subsets has empty boundary?
- ZW-BOUNDARY-OF-A-SET-008 In usual ℝ, what is the boundary of (0,1)∪(1,2)?
- ZW-BOUNDARY-OF-A-SET-009 In a space with the trivial topology \∅,X\, what is the boundary of a nonempty proper subset A?