Closed or open question about the Extended Real line

[Fractal20]

This is just a quick question about sets that include plus or minus infinity on the extended real line. I am wondering about this in regards to measure in analysis so specifically, is [-∞,a) open or closed? I hadn't seen the extended reals before this class and we really didn't spend anytime discussing them. I want to say it is open since it certainly does not have a boundary point on the a) side and it seems like there isn't anything to approach the boundary on the [-∞. Moreover, it's complement [a,∞] looks suspiciously like a closed set... Any verification on these musings? Thanks

 

[Ocifer]

This link should help. Go to Section 6 - Extended Reals, see the 2nd paragraph.

http://math.rice.edu/~semmes/math443.pdf

Also, keep in mind that a set may be neither open nor closed. It might be interesting to find a subset of the extended reals that is neither open nor closed to see how the definitions/requirements given may fail.

 

[Fractal20]

So then it's not open since it isn't open in respect to both τ₊ and τ_-?

 

[mfb]

So then it's not open since it isn't open in respect to both τ₊ and τ_-?

How do you come to that conclusion?
It is wrong, unless you use some non-standard topology.

Also, keep in mind that a set may be neither open nor closed.

Or both at the same time.

 

[Fractal20]

Ah, well now I think it is open. I misunderstood the bit about τ+- in the link. Am I wrong?