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$F_{\sigma}$ denotes a countable union of closed sets and $G_{\delta}$ denotes a countable intersection of open sets.

I can see that there is a different use of article for them. For instance, every half-open interval $[a,b)$ is 'a' $G_{\delta}$ and 'an' $F_{\sigma}$.

Not only this, since i'm a self-studying mathematics, there are some difficulties with notions. Is there a website for notions of mathematics? For example, I can see that some latin word similar to M (I don't know how to type this in Latex) is generally used to denote $\sigma$-algebra on a set $X$, but i have no idea what it is called.. Should i just read it as 'M'? Moreover, i really want to know how other people write latin words. For instance, when you write 'n', you start drawing from the left top, not the right top.

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Read Walter Rudin's Real and Complex Analysis and in the first chapter you'll get that. – Frank Science Jan 7 at 11:51
@Frank I'm reading Rudin's book right now. It just says, "Notation is due to Hausdorff and $\sigma$ refers to the union and $\delta$ to intersection. Where does he say what they are called? I don't understand why Rudin call it " 'a' $G_{\delta}$", but " 'an' $F_{\sigma}$". – Katlus Jan 7 at 12:01
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@Katlus, it is because in English words that start with vowels are prefixed by the article "an" and words that start with consonants are prefixed by the article "a". The letter G is pronounced like "gee" (a soft "g", same as "jee") and F is pronounced like "eff". Since "eff" starts with a vowel, one would say "an eff". – Dan Brumleve Jan 7 at 12:06
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It's worth noting that the a/an distinction is due to phonemic vowels rather than orthographic ones. For example although you say "an umbrella", you say "a union", because "union" has a hidden "y" sound at the front of it. On another note, I wonder if these questions are better suited to Linguistics SE. – Clive Newstead Jan 7 at 12:09
@Dan,Martin Exactly.. Thank you. However i didn't ask why there is a use of 'an' to designate a singular noun starts with a vowel instead of 'a'. See my comment below. – Katlus Jan 7 at 12:11
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You can say a $G_\delta$ as "a gee sub delta" and an $F_\sigma$ as "an eff sub sigma". You can also do without the "sub" if the context is established or if there is a visual reference. $\mathscr{M}$ can be pronounced the same as "M": it is the same letter written in a calligraphic font.

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I couldn't even think this is the reason why.. I thought there is a name for $F_{\delta}$, which starts with a vowel. Thank you – Katlus Jan 7 at 12:09
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@Katlus: The reason for using those letters: $F$ is for French fermé (closed), $G$ is for German Gebiet (zone, region, used for open sets). The $\sigma$ is the Greek "s" for sum (old term for union) and the $\delta$ is the Greek "d" for Durchschnitt (intersection). Hausdorff chose those letters in his Grundzüge der Mengenlehre (1914). For systems of sets $\mathcal{M}$ used a subscript "s" for finite unions from $\mathcal{M}$ and "d" for finite intersections; the Greek counterparts for countable unions/intersections. – Martin Jan 7 at 12:17
@Martin, great info! – Dan Brumleve Jan 7 at 12:22
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On mathoverflow there is What does the $\sigma$ in $\sigma$-algebra stand for? with references. In the comments there are some links and scans. – Martin Jan 7 at 12:41

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