# How are an $ω$-sequence of points and its limit defined?

From Wikipedia:

If the space $$X$$ is sequential, we may say that $$x ∈ X$$ is a limit point of a subset $$S$$ if and only if there is an $$ω$$-sequence of points in $$S - \{x\}$$ whose limit is $$x$$; hence, $$x$$ is called a limit point.

I was wondering how an $$ω$$-sequence of points is defined? Is $$\omega$$ a special ordinal?

How is the limit of an $$ω$$-sequence of points defined?

Thanks and regards!

• Yes, it is the first infinite ordinal. This means that the indices range over all ordinals $<\omega$, that is, the non-negative integers. – André Nicolas Jan 31 '12 at 5:30
• Thanks! Then how is its limit defined? – Tim Jan 31 '12 at 5:31
• $\omega$ represents the set of natural numbers $\mathbb N$. A $\omega$-sequence is just a fancy name for a usual sequence. – azarel Jan 31 '12 at 5:32
• @azarel: Do you mean $\omega \equiv \mathbb{N}$? So a $\omega$-indexed sequence is a $\mathbb{N}$-indexed sequence, i.e. a sequence in the usual sense? – Tim Jan 31 '12 at 5:34
• Tim, yes. $\omega$ sequence is a "regular" sequence. We make the distinction since often we want to allow longer sequences as well. – Asaf Karagila Jan 31 '12 at 5:39

Yes, $\omega$ is the first infinite ordinal. This means that the indices range over all ordinals $<\omega$, that is, the non-negative integers, which are often defined as the finite ordinals.
A similar definition would work for any ordinal $\lambda$. The indices then range over all ordinals $<\lambda$.
For example, if $n$ is a non-negative integer, viewed as an ordinal, then an $n$-sequence has indices ranging over the $n$ (informal) ordinals $<n$, that is, over the ordinals $0,1,\dots,n-1$.
But we could let $\lambda$ be an ordinal $>\omega$. It seems to me you are unlikely to bump into them soon. You can view $\omega$-sequence as an excessively fancy name for an ordinary infinite sequence in the usual sense.