Q1: Let $(X,\mathcal{T})$ be the discrete topological space. Is it a manifold?

A1: Yes, always!

A2: No, never!

A3: There are examples and counterexamples.

Q2: Let $(\mathbb{R},\mathcal{T})$ be the topological space given by the discrete topology. Is it a manifold?

A1: Yes!

A2: No!

Q3: Let $(\mathbb{N},\mathcal{T})$ be the topological space given by the discrete topology. Is it a manifold?

A1: Yes!

A2: No!

A3: There is not enough information to answer this question.

Q4: Let $(\mathbb{R},\mathcal{T})$ be the topological space given by the standard topology. This is a manifold of dimension

A1: $0$.

A2: $1$.

A3: $2$.

A4: $3$.

A5: $4$.

Q5: Let $(M,\mathcal{T})$ be a manifold and $(U_i, h_i)_{i \in I}$ an atlas. What is correct?

A1: $\bigcup_{i \in I} U_i = \emptyset$

A2: $\bigcap_{i \in I} U_i = \emptyset$

A3: $\bigcup_{i \in I} U_i = M$

A4: $\bigcap_{i \in I} U_i = M$