In X=\a,b,c\ let T_1=\∅,X,\a\\ and T_2=\∅,X,\b\\. The union T_1cupT_2 is not a topology. Which set is missing?

Exercise · ZW-TOPOLOGY-006 · ★★☆ Intermediate

In \(X=\{a,b,c\}\) let \(\mathcal{T}_1=\{\varnothing,X,\{a\}\}\) and \(\mathcal{T}_2=\{\varnothing,X,\{b\}\}\). The union \(\mathcal{T}_1\cup\mathcal{T}_2\) is not a topology. Which set is missing?

Choose one option.

Show answer

C. \(\{a,b\}\)

Explanation

\(\{a\}\cup\{b\}=\{a,b\}\) is a union of members of the family that is not in it. Adding it gives a topology.

Study General topology in ZWMath →

Review the concept: Topology

A topology on a set \(X\) is a family \(\mathcal{T}\) of subsets of \(X\), whose members \(U\) are called open sets, containing \(\varnothing\) and \(X\) and closed under arbitrary unions and finite intersections.

\[\varnothing,X\in\mathcal{T};\quad U_i\in\mathcal{T}\Rightarrow\bigcup_i U_i\in\mathcal{T};\quad U_1\cap U_2\in\mathcal{T}\]

In \(X=\{a,b,c\}\) let \(\mathcal{T}_1=\{\varnothing,X,\{a\}\}\) and \(\mathcal{T}_2=\{\varnothing,X,\{b\}\}\). The union \(\mathcal{T}_1\cup\mathcal{T}_2\) is not a topology. Which set is missing?

Explanation

Review the concept: Topology

More practice problems: topology