Describe an open cover that has no finite subcover

Question

Let $S=(1,2)$; describe an open cover of S that has no finite subcover.

Definitions

If $E$ is a set, and {$G$$_\alpha$}$_{\alpha\in A}$ is a collection of sets such that $E \subset \bigcup_{\alpha\in A}$$G$$_\alpha$, then the collection {$G$$_\alpha$}$_{\alpha\in A}$ is called a cover of $E$.

If {$G$$_\alpha$}$_{\alpha\in A}$ is a cover of $E$ and $B\subset A$ such that $E \subset \bigcup_{\alpha\in B}$$G$$_\alpha$ then the collection {$G$$_\alpha$}$_{\alpha\in B}$ is called a subcover of $E$.

Answer 1

$\{\;(1,2-2^{-n})\;:n\in \mathbb N\}.$

Answer 2

'nother

$\{(1.5 - \epsilon, 1.5 + \epsilon)|0< \epsilon < .5\}$. That's an uncountable cover (that does have a countable subcover; e.g. restricting $\epsilon$ to rationals).

Or

$\{(1+\frac{1}{k+1}, 1 + \frac 1{k-1})|k \in \mathbb N; k \ge 2\} = \{(1\frac 13, 2),(1\frac 14, 1\frac 12),(1\frac 15,1 \frac 13),....\}$

or another $\{(r - \epsilon, r + \epsilon)| 1 < r < 2; \epsilon =\frac{\min(r-1, 2 -r)}2 = \text {half the distance from r to the nearest endpoint of (1,2)}\}$

In both of these, if there were a finite subcover, there would be a smallest (yet positive) $\epsilon$ or an largest (yet finite) $k$ so that $(1, 1.5-\epsilon]\cup [1.5+ \epsilon)$ or $(1, \frac 1{k+1}]$ are not covered.

(Similarly in the third, a finite subcover is there'd be a lowest and highest $r$ so there will always be a gap between the lowest or highest point covered and the "edge". That should give you an intuition that, in general, that a compact set must be closed. For a limit point not in the set there are an infinite number of open sets that can get close to the limit point without including it without there being a closest such set.)

(So to come up with a counter example, just think of a sequence that converges to the point and the the open cover is all the open sets "beyond" the point. Ex. $\{(1 + \epsilon, \infty)\}$ or $\{(-\infty,2 -\epsilon)\}$ etc.)

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The first the subcovers are all "nested" so that the you can remove any of the "early ones". Intuitively only the "last" one matters but because there is no last one there can not be a finite subcover.

The second one is an interlocked chain of increasingly small intervals. If you remove any one though you no longer have a cover as each interval contains one point (the upper and lower endpoint of the other two intervals "next to" it) that is not contained in any other.