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Chapter 7
Images and preimages
7.1. Images.
7.1.1. Proposition. Let f : L → M be a localic map and let S ⊆ L be a sublocale.
Then f[S] is a sublocale of M and we have
νf[S](x) = νf[S](y) iff νS(f∗(x)) = νS(f∗(y)). (7.1.1)
Proof. Recall the factorization from 4.6
L f //M
S g∗ with g∗ one-one, hence the formula.
OO jS
OO
jf[S]
// f[S] = fj[S]
with f[S] = fj[S] a sublocale, and with g an onto map (defined by g(x) = f(x)).
S
Thus, the left adjoints form the commutative square
oo f∗ LM
νf[S]
f[S]
νS
oo
g
The sublocale f[S] will be referred to as 43
