Choose the correct statement regarding $\Bbb R$ with the cofinite topology

general-topology

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 $A$.

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