let $(\mathbb{R},\tau)$ be a topological space with the cofinite topology. Every infinite subset of $\mathbb{R}$ is
choose the correct statement
a) Compact but not connecetd
b) Both compact and connected
c)Not compact but connected
d) neither compact nor connected
My attempts : i take $\mathbb{Q}$ subset of $\mathbb{R}$ then it is neither compact not connected so option d) will correct
is it true ??
any hints/solution will be appreciated
thanks u
Best Answer
Hint: For this problem, you need to depend a lot more on the definitions. Your intuition regarding the compact/connected subsets of $\Bbb R$ with the usual topology is not necessarily applicable.
In the co-finite topology, a set is closed if and only if it is finite.
A useful definition of connected is as follows: $A \subset \Bbb R$ will be disconnected (in the cofinite topology) if and only if it can be written as a union $A = B \cup C$, where both $B$ and $C$ are relatively closed.
Compactness in this context is tricky, but remember the definition: a subset $A$ is compact if and only if every open cover of $A$ has a finite subcover. However: if $\mathcal O = \{U_{\alpha}: \alpha \in I\}$ is an open cover of $A$, then any particular open set $U_{\alpha}$ will exclude at most finitely many elements of $A$.