Exercise · ZW-DISCRETE-AND-TRIVIAL-TOPOLOGY-008 · ★★★ Advanced
Let \(\mathcal{T}=\{\varnothing,X,\{1\},\{1,2\}\}\) on \(X=\{1,2,3\}\). How many sets are open in the discrete topology but not in \(\mathcal{T}\)?
Choose one option.
A \(8\) B \(4\) C \(2\) D \(5\) Show answer ← Back to the question
Explanation The discrete topology has 8 open sets and \(\mathcal{T}\) has 4, all of them also open in the discrete topology . That leaves \(8-4=4\).
The discrete topology \(\mathcal{T}_{disc}\) on a set \(X\) has the power set \(\mathcal{P}(X)\) as open sets ; the trivial one \(\mathcal{T}_{triv}\), only the empty set and \(X\).
\[\mathcal{T}_{disc}=\mathcal{P}(X),\qquad \mathcal{T}_{triv}=\{\varnothing,X\}\]
More practice problems: discrete and trivial topology ZW-DISCRETE-AND-TRIVIAL-TOPOLOGY-001 If X has 4 elements, how many open sets does the discrete topology on X have?ZW-DISCRETE-AND-TRIVIAL-TOPOLOGY-002 If X has 4 elements, how many open sets does the trivial topology on X have?ZW-DISCRETE-AND-TRIVIAL-TOPOLOGY-003 Let X=\a,b,c\. In which topology is the set \b\ open?ZW-DISCRETE-AND-TRIVIAL-TOPOLOGY-004 On X=\1,2,3\, which family is the discrete topology?ZW-DISCRETE-AND-TRIVIAL-TOPOLOGY-005 How many topologies on \1,2,3\ are neither the discrete nor the trivial one?ZW-DISCRETE-AND-TRIVIAL-TOPOLOGY-006 In the discrete topology on \1,2,3\, how many open sets contain the point 1?ZW-DISCRETE-AND-TRIVIAL-TOPOLOGY-007 In the trivial topology on \1,2,3\, how many open sets contain the point 1?ZW-DISCRETE-AND-TRIVIAL-TOPOLOGY-009 Let T be any topology on X. Which relation always holds?