Let A be a subset of R with more than one element, let a in A. If A-{a} is compact.

Question

Let $A$ be a subset of $\mathbb R$ with more than one element, let $a$ in $A$. If $A-\{a\}$ is compact, then

1. $A$ is compact
2. Every subset of $A$ must be compact
3. $A$ must be finite set
4. $A$ is disconnected

for any open cover of $A$ is also open cover for $A-\{a\}$, since $A-\{a\}$ is compact implies that it has finite sub cover to cover $A-\{a\}$. with this sub cover union an open set containing $a$ will be the finite sub cover for $A$. Hence $A$ is compact

Given that $A-\{a\}$ is compact, so it is closed, this implies $\{a\}$ is open. But every singleton in $\mathbb{R}$ is closed, so $\{a\}$ is both closed and open. Hence A is disconnected

so option (1) and (4) is right. what can we say about option (2) and (3)?

Answer 1

The proof that 1) is true is ok.

But if $A=\{a\}$ then $A\setminus \{a\}$ is compact and $A$ is connected. So 4) is false.

Counterexamples to 2) and 3) are given in comments: $A=[0,1]\cup\{3\}$ and $a=3$