Exercise · ZW-INTERIOR-OF-A-SET-007 · ★★☆ Intermediate
Which statement holds for every subset \(A\) of any topological space?
Choose one option.
Show answer
- C. \(\operatorname{int}A\subseteq A\).
Explanation
The interior is a union of open sets contained in \(A\), so it lies inside \(A\). Equality holds only when \(A\) is open.
Review the concept: Interior of a set
The interior \(\operatorname{int}A\) of a subset \(A\) of a topological space \((X,\mathcal{T})\) is the largest open set \(U\in\mathcal{T}\) contained in \(A\).
\[\operatorname{int}A=\bigcup\{U\in\mathcal{T}:\ U\subseteq A\}\]
More practice problems: interior of a set
- ZW-INTERIOR-OF-A-SET-001 In X=\1,2,3\ with T=\∅,\1\,\1,2\,\1,2,3\\, what is the interior of A=\1,3\?
- ZW-INTERIOR-OF-A-SET-002 In X=\1,2,3\ with T=\∅,\1\,\1,2\,\1,2,3\\, what is the interior of A=\2,3\?
- ZW-INTERIOR-OF-A-SET-003 In X=\a,b,c,d\ with T=\∅,\a\,\b\,\a,b\,\a,b,c\,\a,b,c,d\\, what is the interior of A=\a,b,d\?
- ZW-INTERIOR-OF-A-SET-004 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 interior of A=\1,2,4…
- ZW-INTERIOR-OF-A-SET-005 In X=\a,b,c\ with T=\∅,\a,b,c\\, what is the interior of A=\a,b\?
- ZW-INTERIOR-OF-A-SET-006 In X=\a,b,c,d\ with T=\∅,\a\,\b\,\a,b\,\a,b,c\,\a,b,c,d\\, how many elements does the interior of A…
- ZW-INTERIOR-OF-A-SET-008 In X=\1,2,3,4\ with T=\∅,\1\,\3\,\1,3\,\1,2,3\,\1,3,4\,\1,2,3,4\\, for which of these sets A does i…
- ZW-INTERIOR-OF-A-SET-009 In X=\1,2,3,4\ with T=\∅,\1\,\3\,\1,3\,\1,2,3\,\1,3,4\,\1,2,3,4\\, how many subsets A⊆ X have empty…