1.3 Sublocals
A nucleus is a map \(e : O(E) \rightarrow O(E)\) with the following three properties:
\(e\) is idempotent
\(U \le e U\)
\(e(U \cap V) = e(U) \cap e(V)\)
(Leroy Lemme 3) Let \(e : O(E) \rightarrow O(E)\) be monotonic. The following are equivalent:
\(e\) is a nucleus
There is a locale \(X\) and a morphism \(f: X \rightarrow E\) such that \(e = f_*f^*\).
Then there is a locale \(X\) and a embedding \(f: X \rightarrow E\) such that \(e = f_*f^*\).
For two nuclei \(e\) and \(f\) on \(O(E)\), we say that \(e \le f\) if \(e(U) \le f(U)\) for all \(U \in O(E)\). This relation is a partial order.
For a set \(S\) of nuclei, the intersection \(\bigcap S\) can be computed by \(\bigcap S(a) = \bigcap \{ j(a) | j \in S \} \). This function satisfies the properties of a nucleus and of an infimum.
Quelle: StoneSpaces S.51
(Leroy CH 3) A sublocal \(Y \subset X\) is defined by a nucleus \(e_Y: O(X) \rightarrow O(X)\), such that \(O(Y) = Im(e_Y) = \{ U \in O(X) | e_Y(U) = U\} \). The corresponding embedding is \(i_X : O(Y) \rightarrow O(X)\). \(i^*_X(V) = e_X(V)\), \((i_X)_*(U) = U\) And every nucleus \(e\) on \(O(X)\) defines a sublocal \(Y\) of \(X\) by \(O(Y) = Im(e)\)
(Stimmt das?)(Leroy Ch 3) \(X \subset Y\) if \(e_Y(u) \le e_X(u)\) for all \(u\). This means that the Sublocals are a dual order to the nuclei.
1.3.1 (1.4) Sublocal Union and Intersection
(Leroy CH 1.4) Let \((X_i)_i\) be a family of sublocals of \(E\) and \((e_i)_i\) the corresponding nuclei. For all \(V \in O(E)\), let \(e(V)\) be the union of all \(W \in O(E)\) which are contained in all \(e_i(V)\).
(Leroy CH 4) Let \(X_i\) be a family of subframes of \(E\) and \(e_i\) be the corresponding nuclei. For every \(V \in O(E)\), let \(e(V)\) be the union of all \(W \in O(E)\) which are contained in every \(e_i(V)\). Then
\(e\) is the corresponding nucleus of a sublocale \(X\) of \(E\)
a sublocale \(Z\) of \(E\) contains \(x\) if and only if it contains all \(X_i\). \(X\) is thus called the union of \(X_i\) denoted by \(\bigcup _i X_i\)
The properties of the nucleus (idempotent, increasing, preserving intersection) can be verified by unfolding the definition of \(e(V)\).
Let \((X_i)_i\) be a family of sublocal of \(E\) and \((e_i)_i\) the corresponding nuclei. For all \(V \in O(E)\), the intersection \(\bigcap X_i\) is the Union of all Nuclei \(w\) such that \(w \le x_i \) for all \(x_i \in X_i \)
The Nuclei (and therefore the sublocals) form a complete lattice.
One can prove that the Nuclei are closed under arbitrary intersections by unfolding the definition of the intersection. The supremum is defined as the infimum of the upper Bound.
A complete Lattice is a Frame if and only if it as a Heyting Algebra.
(Source Johnstone:) The Heyting implication is right adjoint to the infimum. This means that the infimum preserves Suprema, since it is a left adjoint.
The Nuclei form a Heyting Algebra.
Quelle Johnstone
The Nuclei form a frame.
1.3.2 (7) Open Sublocals
Let \(E\) be a space with \(U, H \in O(E)\). We donote by \(e_U\) the largest \(W \in O(E)\) such that \(W \cap U \subset H\). We verify that \(e_U\) is the nucleus of a subspace, which we will temporarily denote by \([U]\).
The map \(e_U\) is a nucleus.
For any \(U \in O(E)\), the sublocal \([U]\) is called an open sublocal of \(E\).
(Leroy Lemma 6,7)
For all subspaces \(X\) of \(E\) and any \(U \in O(E)\):
\[ X \subset [U] \iff e_X(U) = 1_E \]For all \(U, V \in O(E)\), we have:
\[ [U \cap V] = [U] \cap [V] \]\[ e_{U \cap V} = e_Ue_V=e_Ve_U \]\[ U \subset V \iff [U] \subset [V] \]For all families \(V_i\) of elements of \(O(E)\), we have:
\[ \cup _i[V_i] = [\cup _iV_i] \]
The complement of an open sublocal \(U\) of \(X\) is the sublocal \(X \setminus U\). (Leroy p. 12)
The complement is injective.
A sublocal \(X\) of \(E\) is called closed if \(X = E \setminus U\) for some open sublocal \(U\) of \(E\).
For any family \(X_i\) of closed sublocals of \(E\), the intersection \(\bigcap X_i\) is closed (it can be computed by taking the complement of the union of the complements).
For any open sublocal \(V\) of \(E\) and any sublocal \(X\) of \(E\), we have:
And thereby:
For any open sublocal \(V\) of \(E\) and any sublocal \(X\) of \(E\), we have:
\(Int X\) is the largest open sublocal contained in \(X\)
\(Ext X\) is the largest open sublocal contained in \(E \setminus X\)
\(\bar{X}\) is the smallest closed sublocal containing \(X\)
\(\partial X = \bar{X} \cap (E - Int X)\)
\(\bar{X} = E \setminus Ext(X)\)
\(\partial X = E \setminus (Int X \cup Ext X)\)
\(Int X \cup \partial X = \bar X\)
\(Ext X \cup \partial X = E \setminus Int X\)