MATH101 - Math101 - QUIZ

Questions

1. 1Let X be a finite set and \(\|\mathcal{T}\|\) be a topology on X. If \(\|\mathcal{T}\'\) is defined as \(\|\mathcal{T}\' = \{S \mid X - S \in \mathcal{T}\} \), which of the following conditions must be met for \(\|\mathcal{T}\'\) to be a topology on X?
2. 2Consider the space \(X = \mathbb{R} \times [0,1]\) with the dictionary order and the order topology \(\mathcal{T}\). Let \(\mathcal{T}^*\) be the topology generated by the basis \(\{(a, b) \times [0, 1] \mid a < b \}\). For \(\mathcal{T}^*\) to be finer than \(\mathcal{T}\), which of the following must be true?
3. 3Which of the following is an example of a subset of \(\mathbb{R}\) that is neither open nor closed in the lower limit topology?
4. 4Consider a group \(G\) such that \(|G| > 2\) and its only subgroups are the trivial subgroup \(\{e\}\) and \(G\) itself. Which of the following is a possible example of such a group?
5. 5Which of the following describes a set \(S\) with an order relation where every element has an immediate successor, but at least one element (other than the smallest) does not have an immediate predecessor?
6. 6Let \(X = \{a,b,c,d\}\). Consider a topology \(\mathcal{T}\) on \(X\) such that \(\{a,b\} \in \mathcal{T}\) and \(\{b,c\} \in \mathcal{T}\). If \(\mathcal{T}\) is not the discrete topology, which of the following is a valid topology \(\mathcal{T}\)?