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I wanted to prove following exercise

$R_l$ lower limit topology is Lindelöf Space.

Lindelöf space is space with every cover has countable cover.

I tried but I am not able to even start.

Please give me hint so that I could complete this .

Any Help will be appreciated

Jean-Claude Arbaut
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Curious student
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1 Answers1

8

Some copious hints:

$1).$ It is enough to show that every open covering of $\mathbb R_{\ell}$ by basis elements has a countable subcover.

$2).$ Let $\mathscr A = \{[a_i, b_i) | i ∈ J\}$ be such a cover and consider the union of intervals $A' = \bigcup_{i\in J} (a_i, b_i).$

$3).$ If $x\in \mathbb R\setminus A'$ then $x\in [a_i,b_i)$ for some $i\in J.$ Choose a rational $r_x\in (a_i,b_i)$ and show that $x<y\Rightarrow r_x<r_y$ and thus that $ \mathbb R\setminus A'$ is countable.

$4).$ For each $x\in \mathbb R\setminus A'$ choose an element of $\mathscr A$ that contains it. The resulting sets form a countable subcollection of $\mathscr A$ that covers $\mathbb R\setminus A'$.

$5).\ A'$ is open in the standard topology on $\mathbb R$ so it has a countable subcover $\{(a_n,b_n)\}$. Then, $\{[a_n,b_n)\}$ is a countable subcover of $A'$, and this, together with $4).$, proves the claim.

Matematleta
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