Exercise · ZW-TOPOLOGICAL-SPACE-009 · ★★★ Advanced
How many topologies on \(\{1,2,3\}\) have exactly 3 open sets?
Choose one option.
Show answer
- B. \(6\)
Explanation
With 3 open sets the family is \(\{\varnothing,X,U\}\) with \(U\) a nonempty proper subset. Any of the 6 such subsets works, and each gives a different topology.
Review the concept: Topological space
A topological space is a set \(X\) together with a topology \(\mathcal{T}\) on it; the members of \(\mathcal{T}\) are its open sets \(U\).
\[(X,\mathcal{T}),\qquad U\text{ open}\iff U\in\mathcal{T}\]
More practice problems: topological space
- ZW-TOPOLOGICAL-SPACE-001 Let X=\1,2,3\ with the topology T=\∅,X,\1\,\1,2\\. Which of these sets is open?
- ZW-TOPOLOGICAL-SPACE-002 Is \∅,X,\1\,\2\\ a topology on X=\1,2,3\?
- ZW-TOPOLOGICAL-SPACE-003 How many open sets does the space (X,T) have, with X=\1,2,3\ and T=\∅,X,\1\,\1,2\\?
- ZW-TOPOLOGICAL-SPACE-004 How many different topologies can be defined on the set \1,2,3\?
- ZW-TOPOLOGICAL-SPACE-005 In ℝ with the usual topology, which of these sets is open?
- ZW-TOPOLOGICAL-SPACE-006 Let Y=\a,b,c\ with T=\∅,Y,\a\,\b\,\a,b\\. Which set is open?
- ZW-TOPOLOGICAL-SPACE-007 In ℕ, let T be the family made of ∅ and the sets whose complement is finite. Which one is open?
- ZW-TOPOLOGICAL-SPACE-008 Among the topologies on \1,2\, in how many is the set \1\ open?