please help me understand what this subcovering question is asking for.

Question

for each $x\in(0,1)$, let $I_x$ denote the open interval $(\frac{x}{2},\frac{(x+1)}{2})$. show that the family $G$ of all such $I_x$ is an open covering of $(0,1)$ which admits no finite subcovering of $(0,1)$
I am studying Heine Borel property. I am no sure but is this question asking to prove $(0,1)$ does not have a Heine-Borel property?
any clue how to begin, what to look for? Thanks.

Answer 1

To show it is an open cover, you need to show that every set in the family $G$ is open and that they cover $(0,1)$. Given a $y \in (0,1)$ you need to show that it is in at least one of the $I_x$. To show it has no finite subcover, assume you are given a finite set of the $I_x$, so $I_{x_1},I_{x_2},I_{x_3},I_{x_4},\ldots I_{x_n}$. Find some $y \in (0,1)$ that is not in any of the intervals.

Answer 2

Yes, the problem is indeed asking you to prove that $(0,1)$ does not have the Heine-Borel property. However, the problem is very nicely giving you a big hint about how to go about it.

The Heine-Borel property for $(0,1)$ says: every open cover of $(0,1)$ has a finite subcover.

So to disprove the Heine-Borel property for $(0,1)$ you must prove: there exists an open cover of $(0,1)$ which has no finite sub cover.

Ordinarily, to carry out this existence proof, you would have to use your imagination or your mathematical experience or whatever you can use, in order to write down an appropriate collection of open subsets; and then you still have to prove that collection is an open cover of $(0,1)$ and that it has no finite subcover.

In this case, however, the problem writer has very kindly handed you an appropriate collection of open subsets, namely $\{I_x \mid x \in (0,1)\}$. But now you still have to prove that $\{I_x \mid x \in (0,1)\}$ is an open cover of $(0,1)$ and that it does not have a finite subcover.

Answer 3

Yes, this exercise shows that $(0,1)$ does not have the Heine-Borel property: there is an open cover without a finite subcover.

Theorem: in $\Bbb R$ in the Euclidean metric, a set has the Heine-Borel property iff it is closed and bounded. This example shows what can go wrong if the set is bounded and not closed.

To start with why it is a cover: can you find an element of the cover that contains $\frac12$, e.g.?