# To prove $\omega_1 \cup \{\omega_1\}$ is not first countable but $\omega_1$ is

The topological space $X=\omega_1\cup \{\omega_1\}$ is not first countable but $\omega_1$ is. Here $\omega_1$ denotes the set of all countable ordinals and $\{\omega_1\}$ is the first uncountable ordinal. Cardinality of $\omega_1$ is $\aleph_1.$And also $\omega_1$ is called the limit ordinal meaning it is the limit point of the set $\omega_1.$

Here I want to prove two things. $1)$ $\omega_1$ is first countable. and $2)$ $\{\omega_1\}$ is the limit point of $\omega_1$. But since no sequence converges to $\{\omega_1\}$, using this I could say that our space $X$ is not first countable.

$1)$ The proof of this I got from here with just one question: it says we need to prove that such an $S(y)$ exists,and I'm thinking $S(y)=\cap\{u\in \omega_1:u\gt y\}.$ This is correct$?$ But here is a chance that the countable base around $y$ may be finite and not countably infinite.

$2)$ If possible let us assume that $\omega_1$ is not a limit point of the set $\omega_1.$ $U$ be a nbd of $\omega_1$ that does not intersect $\omega_1.$ Then what are the elements of $U$ like? The ones coming before $\omega_1$ cannot be uncountable because $\omega_1$ is the first. Then they must be countable. This is absurd so no such $U$ exists. Thus $\{\omega_1\}$ is a limit point.

Are my proofs correct? Thank you.

• In "first countable" and "second countable", we include finite in the definition of countable. Otherwise no finite space is second countable, and the theorem that a compact space is metrizable if and only if it is second countable fails. – Asaf Karagila Dec 27 '16 at 6:10
• @AsafKaragila: what? – user80631 Dec 27 '16 at 6:36
• To your first question. Finite sets are countable, at least in the context of the definition in "first countable". – Asaf Karagila Dec 27 '16 at 6:36
• You should not really be considering limits. A space $S$ may have the property that whenever $p\in S$ and $T\subset S$ and $p\in \bar S,$ there is a countable sequence in $T$ converging to $p,$ and yet $X$ may fail to be first-countable. Example: The box-product topology on $\mathbb R^{\omega}.$ – DanielWainfleet Dec 27 '16 at 22:55

A neighbourhood base for an ordinal $\alpha \in \omega_1$, either $\{\alpha\}$, when $\alpha$ is an isolated point, which happens when it is a successor ordinal or $0$ : if $\alpha=\beta+1$, then $\{\alpha\}=(\beta, \alpha+1)$,so open as an open interval.$0$ being the minimal element has $[0,1)=\{0\}$ as an open neighbourhood.Isolated points have a finite local base, which is no problem as first countable means every point has a local base that is at most countable, it need not be infinite per se.

If $\alpha \in \omega$ is a limit ordinal, it also has a local base that is countable, namely all sets of the form $(\beta, \alpha+1)$, where $\beta < \alpha$, of which there are but countably many as $\alpha$ is a countable ordinal.

If $U_n$ were a countable base at $\omega_1 \in \omega_1 +1$, then as it is the max of the set, every $U_n$ must contain (by the definition of order topology) a set of the form $(\alpha_n, \omega_1]$, where $\alpha_n$ is some countable ordinal. BUt then $\alpha =\cup_n \alpha_n$ is also a countable ordinal (a countable union of countable ordinals), which is the sup of the $\alpha_n$. But then $(\alpha+1, \omega_1]$ is an open neighbourhood of $\omega_1$ that contains no $U_n$.

So no countable family of open neighbourhoods of $\omega_1$ can form a local base.

• ok. thanks for the answer but can we show that $\omega_1$ is a limit point of $\omega_1\ ?$ – user80631 Jan 1 '17 at 21:15
• It's.a max so neighbourhoods are final tails. – Henno Brandsma Jan 1 '17 at 21:45
• What is a final tail? – roi_saumon Dec 5 '20 at 15:08
• @roi_saumon s set like $(\alpha, \omega_1]$. – Henno Brandsma Dec 5 '20 at 15:47

(1) Your construction of $S(y)$ is correct (though of course you still need to actually prove that it works). A finite set is countable, so there is no issue with having a finite local base instead of a countably infinite local base. That is, a space is first-countable if every point has a countable local base, which includes the possibility that the local base is finite.

(2) Your argument shows that if $U$ is a neighborhood of $\omega_1$ that does not intersect $\omega_1$, then $U$ contains no ordinals less than $\omega_1$. But you have not explained why this gives a contradiction. Maybe the singleton set $\{\omega_1\}$ is open in $X$.

• That's right. The singleton $\{\omega_1\}$ could be open. But how do we show that it's not.not . But the ordinals are in order topology no? The basic open sets are of form $(a,b), [a,b), (a,b]$ where the second and third cases are respectively for $a$ being the smallest and $b$ being the largest element of the set. But in this $\omega_1$ is neither. Will this do ? – user80631 Dec 27 '16 at 9:44