Exercise · ZW-DISCRETE-AND-TRIVIAL-TOPOLOGY-004 · ★☆☆ Basic
On \(X=\{1,2,3\}\), which family is the discrete topology?
Choose one option.
Show answer
- C. all \(8\) subsets of \(X\)
Explanation
The discrete topology is the whole power set. Removing \(\varnothing\) breaks the first axiom, and the family \(\{\varnothing,X,\{1\},\{2\},\{3\}\}\) lacks \(\{1,2\}\), a union of two open sets.
Review the concept: Discrete and trivial topology
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-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-008 Let T=\∅,X,\1\,\1,2\\ on X=\1,2,3\. How many sets are open in the discrete topology but not in T?
- ZW-DISCRETE-AND-TRIVIAL-TOPOLOGY-009 Let T be any topology on X. Which relation always holds?
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