# Extended real line is 2nd Countable (Clarification).

The extended real line can be given the order topology i.e the topology generated by the basis elements $$(a,b)=\{aa\}$$ and $$L_a=\{x where $$a,b\in \bar R$$.

Hence, a countable basis would just be the above sets except with the restriction that $$a,b\in \mathbb Q\cup \{+\infty,-\infty\}$$

• Well the real line is second countable? It's just not if you use the lower limit topology. You can check the wikipedia article. en.wikipedia.org/wiki/Second-countable_space – 0CT0 Aug 29 '19 at 15:27
• @0CT0 An extension of a second countable space need not always be second countable. So it's a reasonable question. – Henno Brandsma Aug 29 '19 at 17:04

Yes, that's true. Restricting the base elements to rational endpoints (in all 3 types) gives a countable base, as Q is order dense in the extended reals: for any $$a < b$$ we can find a rational strictly in-between.
As all order topologies are $$T_3$$ (even $$T_5$$) Urysohn's metrication theorem already tells us it's metrisable (which is also obvious if you know that $$\bar{\Bbb R}$$ is homeomorphic to $$[-1,1] \subseteq \Bbb R$$.)
• Hi just to clarify the points need only come from $\mathbb Q$ and not necessarily $\mathbb Q\cup \{+\infty,-\infty \}$? – Jhon Doe Aug 30 '19 at 10:48
• @JhonDoe no $\Bbb Q$ more than suffices. – Henno Brandsma Aug 30 '19 at 10:52