Lemma 5.19.8. Suppose that $s, t : R \to U$ and $\pi : U \to X$ are continuous maps of topological spaces such that

1. $\pi$ is open,

2. $U$ is sober,

3. $s, t$ have finite fibres,

4. generalizations lift along $s, t$,

5. $(t, s)(R) \subset U \times U$ is an equivalence relation on $U$ and $X$ is the quotient of $U$ by this equivalence relation (as a set).

Then $X$ is Kolmogorov.

Proof. Properties (3) and (5) imply that a point $x$ corresponds to an finite equivalence class $\{ u_1, \ldots , u_ n\} \subset U$ of the equivalence relation. Suppose that $x' \in X$ is a second point corresponding to the equivalence class $\{ u'_1, \ldots , u'_ m\} \subset U$. Suppose that $u_ i \leadsto u'_ j$ for some $i, j$. Then for any $r' \in R$ with $s(r') = u'_ j$ by (4) we can find $r \leadsto r'$ with $s(r) = u_ i$. Hence $t(r) \leadsto t(r')$. Since $\{ u'_1, \ldots , u'_ m\} = t(s^{-1}(\{ u'_ j\} ))$ we conclude that every element of $\{ u'_1, \ldots , u'_ m\}$ is the specialization of an element of $\{ u_1, \ldots , u_ n\}$. Thus $\overline{\{ u_1\} } \cup \ldots \cup \overline{\{ u_ n\} }$ is a union of equivalence classes, hence of the form $\pi ^{-1}(Z)$ for some subset $Z \subset X$. By (1) we see that $Z$ is closed in $X$ and in fact $Z = \overline{\{ x\} }$ because $\pi (\overline{\{ u_ i\} }) \subset \overline{\{ x\} }$ for each $i$. In other words, $x \leadsto x'$ if and only if some lift of $x$ in $U$ specializes to some lift of $x'$ in $U$, if and only if every lift of $x'$ in $U$ is a specialization of some lift of $x$ in $U$.

Suppose that both $x \leadsto x'$ and $x' \leadsto x$. Say $x$ corresponds to $\{ u_1, \ldots , u_ n\}$ and $x'$ corresponds to $\{ u'_1, \ldots , u'_ m\}$ as above. Then, by the results of the preceding paragraph, we can find a sequence

$\ldots \leadsto u'_{j_3} \leadsto u_{i_3} \leadsto u'_{j_2} \leadsto u_{i_2} \leadsto u'_{j_1} \leadsto u_{i_1}$

which must repeat, hence by (2) we conclude that $\{ u_1, \ldots , u_ n\} = \{ u'_1, \ldots , u'_ m\}$, i.e., $x = x'$. Thus $X$ is Kolmogorov. $\square$

Comment #742 by Wei Xu on

Typos, "$t(t)$" in "Hence $t(t) \leadsto t(r')$" should be "$t(r)$".

And,

"specializes" in

"In other words, $x \leadsto x'$ if and only if some lift of $x$ in $U$ specializes to some lift of $x'$ in $U$" should be "generalizes".

Comment #6771 by Alejandro González Nevado on

At the end, shouldn't it be say that there is not only a sequence but several sequences (one for each $u_{i}$ beginning on such $u_{i}$)? If there was only a sequence, then we would be saying that all elements are in such sequence and thence proving that $u_{i}=u_{j}=u'{k}=u'_{l}$, i.e., that all the elements at play are equal and not the sets $\{u_{1},\dots,u_{n}\}$ and $\{u'_{1},\dots,u'_{m}\}$ themselves.

Comment #6773 by on

Once the sets $\{u_i\}$ and $\{u'_j\}$ have one element in comment they are the same.

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