Questions on topology [closed]

Question

Q 1. In $\mathbb R$ with the finite complement topology if $U$ is open then $U$ is uncountable. Is it true or false? Give example to support your answer.

Q 2. Let $X$ be a set and let $p$ belongs to $X$. Define a collection of sets $T = \{ U \subset X : p \in U \mbox{ or }U =\emptyset \}$. Then $T$ is a topology on $X$. Is it true or false? Give example to support your answer.

Q 3. Let $( X, T)$ be a topological space and let $S$ be a subspace of $X$. If $D$ is closed in $X$, then $S\cap D$ is closed in $S$. Is it true or false?

Answer 1

Q1. True for $U \neq \varnothing$. Because if $U$ is open and non empty then the complement of $U$ is finite hence the cardinality of $U$ must be continuum, i.e. uncountable. Example $U=\mathbb{R} \setminus \lbrace 1,2 \rbrace$ is uncountable.

Q2. True. Because given two open sets $U_1$ and $U_2$, then if one of them is empty, their intersection is empty. Otherwise, both must contain $p$ hence their intersection contains $p$, so their intersection is open. Similarly given any family of open sets its union is open. Finally $p \in X$ and the empty set are both open so this defines a topology.

Q3. True.