Exercise · ZW-DISCRETE-AND-TRIVIAL-TOPOLOGY-009 · ★★☆ Intermediate
Let \(\mathcal{T}\) be any topology on \(X\). Which relation always holds?
Choose one option.
Show answer
- C. \(\mathcal{T}_{triv}\subseteq\mathcal{T}\subseteq\mathcal{T}_{disc}\)
Explanation
Every topology contains \(\varnothing\) and \(X\), so it contains the trivial one, and its open sets are subsets of \(X\), so it lies inside the discrete one.
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-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-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?
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