(Some small parts of the proof from [11] are applied in the proof).

First note that \(G_{\delta}\)-sets of \(X\) form a base for open sets of \(X_{\delta}\). Thus, it is enough to prove that there are at most \(2^{c{(X)}}\) disjoint \(G_{\delta}\)-sets in \(X_{\delta}\).

Suppose in contrary that \({c{(X_{\delta})}} > 2^{c{(X)}}\). Observe that every non-empty \(G_{\delta}\)-set of \(X\) contains a non-empty closed \(G_{\delta}\)-set. Indeed. Choose \(x \in X\) arbitrarily and take the family \(\{ G_{n}:{n \in \omega}\}\) of open sets such that \(x \in {\bigcap_{n \in \omega}G_{n}}\). Next, for each \(n \in \omega\) consider open neigbourhoods \(\{ U_{n,m}:{m \in \omega}\}\) of \(x\) such that \(\overline{U_{n,1}} \in G_{n}\). It is possible because \(X\) is regular. Next, choose open sets \(U_{n,m} \subset X\) such that \({x \in U_{n,m} \subset \overline{U_{n,m}} \subset U_{n,{m - 1}}}.\) Define \(G = {\bigcap{\{ U_{n,m}:{{n,m} \in \omega}\}}}\). Since

we conclude that \(G\) is closed and we have that \(G \subset {\bigcap_{n \in \omega}G_{n}}\). Hence, we will restrict our considerations to examing families of closed \(G_{\delta}\)-sets. We admit the fact as well-known that in a compact Hausdorff space every closed \(G_{\delta}\)-set has a countable neighbourhood base.

Now, we will construct an \(\omega^{+}\)-strong sequence \(\{{({\{ V_{\beta}\}},\mathcal{H}_{\beta})}:{\beta < {(2^{c{(X)}})}^{+}}\}\) of the following properties:

• (1)

  \(V_{\beta}\) is a \(G_{\delta}\)-set in \(X\);

• (2)

  \(\mathcal{H}_{\beta}\) is a (countable) neighbourhood base for \(V_{\beta}\) in \(X\);

• (3)

  \(V_{\beta} \in {{P{(X)}} \smallsetminus {\{\mathcal{H}_{\alpha}:{\alpha < \beta}\}}}\);

• (4)

  \(\exists_{{V,W} \subset X}\) \(V_{\beta} \subset V\) and \({\bigcup{\{ V_{\alpha}:{\alpha < \beta}\}}} \subset W\) and \({V \cap W} = \varnothing\);

• (5)

  \({({\{ V_{\beta}\}},\mathcal{H}_{\beta})} = {({\{{\bigcup_{\alpha < \beta}V_{\alpha}}\}},{\bigcup_{\alpha < \beta}\mathcal{H}_{\alpha}})}\) for limit \(\beta\).

We start the construction with choosing an arbitrary \(G_{\delta}\)-set \(V_{0} \subsetneq X\). Let \(\mathcal{H}_{0}\) be a neighbourhood base for \(V_{0}\) in \(X\). By the well-known fact mentioned above \({|\mathcal{H}_{0}|} \leqslant \omega\). Let \(({\{ V_{0}\}},\mathcal{H}_{0})\) be the first pair of an \(\omega^{+}\)-strong sequence.

Assume that for some \(\alpha < \beta < {(2^{c{(X)}})}^{+}\) we have constructed the \(\omega^{+}\)-strong sequence

of properties \({(1)} - {(5)}\). Since the limit step is done, we will construct \(\beta = {({\alpha + 1})}\)-step.

Observe that for each \(\alpha < \beta < {(2^{c{(X)}})}^{+}\) we have that \({|\mathcal{H}_{\alpha}|} \leqslant \omega\). Hence

Hence \({{P{(X)}} \smallsetminus {\{\mathcal{H}_{\alpha}:{\alpha < \beta}\}}} \neq \varnothing\). Let \(V_{\beta} \in {{P{(X)}} \smallsetminus {\{\mathcal{H}_{\alpha}:{\alpha < \beta}\}}}\) be an arbitrary \(G_{\delta}\)-set in \(X\). Since \(X\) is normal there are open and disjoint sets \(U_{\alpha},U_{\beta}\) such that \(V_{\beta} \subset U_{\beta}\) and \({\bigcup{\{ V_{\alpha}:{\alpha < \beta}\}}} \subset U_{\alpha}\). Let \(\mathcal{H}_{\beta}\) be a neighbourhood base for \(V_{\beta}\) in \(X\). Let \(({\{ V_{\beta}\}},\mathcal{H}_{\beta})\) be the next pair of the \(\omega^{+}\)-strong sequence.

Thus, we have defined the \(\omega^{+}\)-strong sequence \(\{{({\{ V_{\beta}\}},\mathcal{H}_{\beta})}:{\beta < {(2^{c{(X)}})}^{+}}\}\) fulfilling properties \({(1)} - {(5)}\). Now, apply Theorem 2.3 for \(({P{(X)}}, \in )\), \(\kappa = \omega^{+}\), \(2^{\mu} = \omega\), \(\eta = {(2^{c{(X)}})}^{+}\) and \(\beta = {({c{(X)}})}^{+}\). By Theorem 2.3 we obtain the \(\omega^{+}\)-strong sequence \(\{{({\{ W_{\beta}\}},\mathcal{H}_{\beta})}:{\beta < {({c{(X)}})}^{+}}\}\) such that \(W_{\beta} \subseteq V_{\beta}\) for each \(\beta < {({c{(X)}})}^{+}\). By our construction \({W_{\alpha} \cap W_{\beta}} = \varnothing\) for distinct \({\alpha,\beta} < {({c{(X)}})}^{+}\). Hence, we obtain \({({c{(X)}})}^{+}\) of pairwise disjoint elements in \(X\). A contradiction.

Suppose that \({\pi w{(X)}} > {({\pi\chi{(X)}})}^{c{(X)}}\). Let \(\mathcal{B}\) be a \(\pi\)-base of cardinality \(\pi w{(X)}\). We will construct a \({({\pi\chi{(X)}})}^{+}\)-strong sequence \(\{{({\{ U_{\beta}\}},\mathcal{W}_{\beta})}:{\beta < {({({\pi\chi{(X)}})}^{c{(X)}})}^{+}}\}\) of the following properties:

• (1)

  \(U_{\beta} \in \mathcal{B}\);

• (2)

  \(x_{\beta} \in U_{\beta}\);

• (3)

  \(\mathcal{W}_{\beta}\) is a local \(\pi\)-base for \(x_{\beta} \in {\mathcal{B} \smallsetminus {\{\mathcal{W}_{\alpha}:{\alpha < \beta}\}}}\);

• (4)

  \({({\{ U_{\beta}\}},\mathcal{W}_{\beta})} = {({\{{\bigcup_{\alpha < \beta}U_{\alpha}}\}},{\bigcup_{\alpha < \beta}\mathcal{W}_{\alpha}})}\) for limit \(\beta\).

Let \(U_{0} \in \mathcal{B}\) be an arbitrary set and let \(x_{0} \in U_{0}\) be an arbitrary element. Let \(\mathcal{W}_{0}\) be a local \(\pi\)-base for \(x_{0}\) in \(X\). Obviosuly \({|\mathcal{W}_{0}|} \leqslant {\pi\chi{(X)}} < {\pi w{(X)}}\) (by our assumption). Let \(({\{ U_{0}\}},\mathcal{W}_{0})\) be the first pair of a \({({\pi\chi{(X)}})}^{+}\)-strong sequence.

Assume that for some \(\alpha < \beta < {({({\pi\chi{(X)}})}^{c{(X)}})}^{+}\) we have constructed the \({({\pi\chi{(X)}})}^{+}\)-strong sequence \(\{{({\{ U_{\alpha}\}},\mathcal{W}_{\alpha})}:{\alpha < \beta}\}\) of properties \({(1)} - {(4)}\). Since the limit step is done, we will construct \(\beta = {({\alpha + 1})}\)-step.

Observe that for each \(\alpha < \beta < {({({\pi\chi{(X)}})}^{c{(X)}})}^{+}\) we have \({|\mathcal{W}_{\alpha}|} \leqslant {\pi\chi{(X)}} < {\pi w{(X)}}\) and hence

Hence \({\mathcal{B} \smallsetminus {\{\mathcal{W}_{\alpha}:{\alpha < \beta}\}}} \neq \varnothing\). Let \(x_{\beta} \in {\mathcal{B} \smallsetminus {\{\mathcal{W}_{\alpha}:{\alpha < \beta}\}}}\). Such choice is possible because of our assumption (and \({\pi w{(X)}} = {{{d{(X)}} \cdot \pi}\chi{(X)}}\), (see [8, 3.8(b)])).

Since \(X\) is almost regular there are disjoint regularly open sets \({W_{\beta},W_{\alpha}} \subset X\) such that \(x_{\beta} \in W_{\beta}\) and \(A_{\alpha} \subset W_{\alpha}\), where \(A_{\alpha}\) is a regularly closed set such that \({\bigcup{\{ x_{\alpha}:{\alpha < \beta}\}}} \subset A_{\alpha}\). Let \(U_{\beta} \subset W_{\beta}\) be a set such that \(x_{\beta} \in U_{\beta} \in \mathcal{B}\). Let \(({\{ U_{\beta}\}},\mathcal{W}_{\beta})\) be the next pair of the \({({\pi\chi{(X)}})}^{+}\)-strong sequence.

Thus, we have defined the \({({\pi\chi{(X)}})}^{+}\)-strong sequence \(\{{({\{ U_{\beta}\}},\mathcal{W}_{\beta})}:{\beta < {({({\pi\chi{(X)}})}^{c{(X)}})}^{+}}\}\) of properties \({(1)} - {(4)}\). Now, apply Theorem 2.3 to \((\mathcal{B}, \in )\), \(\kappa = 2^{\mu} = {({\pi\chi{(X)}})}^{+}\), \(\beta = {({c{(X)}})}^{+}\) and \(\eta = {({({\pi\chi{(X)}})}^{c{(X)}})}^{+}\). By Theorem 2.3 we obtain a \({({\pi\chi{(X)}})}^{+}\)-strong sequence \({{\{{({\{ V_{\beta}\}},\mathcal{W}_{\beta})}:\beta < )}c{(X)})}^{+}\}\) such that \(V_{\beta} \subseteq U_{\beta}\) for each \(\beta < {({c{(X)}})}^{+}\). By our construction \({V_{\alpha} \cap V_{\beta}} = \varnothing\) for distinct \({\alpha,\beta} < {({c{(X)}})}^{+}\). Hence, we obtain \({({c{(X)}})}^{+}\) of pairwise disjoint elements in \(X\). A contradiction.

Suppose that \({|X|} > 2^{\chi{(X)}wL{(X)}}\). We will construct a \({({wL{(X)}})}^{+}\)-strong sequence \(\{{({\{ U_{\beta}\}},\mathcal{C}_{\beta})}:{\beta < {(2^{\chi{(X)}wL{(X)}})}^{+}}\}\) of the following properties

• (1)

  \(\mathcal{C}_{\beta} \subset {{P{(X)}} \smallsetminus {\{\mathcal{C}_{\alpha}:{\alpha < \beta}\}}}\) is a family such that \({|\mathcal{C}_{\beta}|} \leqslant {wL{(X)}}\) and \(\overline{\bigcup\mathcal{C}_{\beta}} = X\);

• (2)

  \(U_{\beta} \in \mathcal{C}_{\beta}\);

• (3)

  \({({\{ U_{\beta}\}},\mathcal{C}_{\beta})} = {({\{{\bigcup_{\alpha < \beta}U_{\alpha}}\}},{\bigcup_{\alpha < \beta}\mathcal{C}_{\alpha}})}\) for limit \(\beta\).

We start the construction with choosing an open cover \(\mathcal{D}_{0}\) of \(X\). Let \(\mathcal{C}_{0}\) be a subfamily of \(\mathcal{D}_{0}\) such that \({|\mathcal{C}_{0}|} \leqslant {wL{(X)}}\) and \(\overline{\bigcup\mathcal{C}_{0}} = X\). Let \(U_{0} \in \mathcal{C}_{0}\) be an arbitrary element. Let \(({\{ U_{0}\}},\mathcal{C}_{0})\) be the first pair of a \({({wL{(X)}})}^{+}\)-strong sequence.

Assume that for some \(\alpha < \beta < {(2^{\chi{(X)}wL{(X)}})}^{+}\) we have constructed the \({({wL{(X)}})}^{+}\)-strong sequence \(\{{({\{ U_{\alpha}\}},\mathcal{C}_{\alpha})}:{\alpha < \beta}\}\) of properties \({(1)} - {(3)}\). Since the limit step is done, we will construct \(\beta = {({\alpha + 1})}\)-step.

Observe that for each \(\alpha < \beta < {(2^{\chi{(X)}wL{(X)}})}^{+}\) we have \({|\mathcal{C}_{\alpha}|} \leqslant {wL{(X)}}\). Hence

Hence \({{P{(X)}} \smallsetminus {\{\mathcal{C}_{\alpha}:{\alpha < \beta}\}}} \neq \varnothing\). Let \(\mathcal{D}_{\beta} \subset {{P{(X)}} \smallsetminus {\{\mathcal{C}_{\alpha}:{\alpha < \beta}\}}}\) be an open cover of \(X\) and let \(\mathcal{C}_{\beta}\) be a subfamily of \(\mathcal{D}_{\beta}\) such that \({|\mathcal{C}_{\beta}|} \leqslant {wL{(X)}}\) and \(\overline{\bigcup\mathcal{C}_{\beta}} = X\). Let \(U_{\beta} \in \mathcal{C}_{\beta}\) be an arbitrary element. Let \(({\{ U_{\beta}\}},\mathcal{C}_{\beta})\) be the next pair of the \({({wL{(X)}})}^{+}\)-strong sequence.

Thus, we have defined the \({({wL{(X)}})}^{+}\)-strong sequence \(\{{({\{ U_{\beta}\}},\mathcal{C}_{\beta})}:{\beta < {(2^{\chi{(X)}wL{(X)}})}^{+}}\}\) of properties \({(1)} - {(3)}\). Now, apply Theorem 2.3 to \(({P{(X)}}, \in )\), \(\kappa = {({wL{(X)}})}^{+}\), \(2^{\mu} = {wL{(X)}}\), \(\beta = {({\chi{(X)}})}^{+}\) and \(\eta = {(2^{\chi{(X)}wL{(X)}})}^{+}\). By Theorem 2.3 we obtain \({({wL{(X)}})}^{+}\)-strong sequence \(\{{({\{ V_{\beta}\}},\mathcal{C}_{\beta})}:{\beta < {({\chi{(X)}})}^{+}}\}\) such that \(V_{\beta} \subset U_{\beta}\) for each \(\beta < {({\chi{(X)}})}^{+}\).

Now, for each \(x \in {\bigcup_{\beta < {({\chi{(X)}})}^{+}}\mathcal{C}_{\beta}}\) we define \({\mathcal{B}{(x)}} = {\{{W_{\beta} \subset \mathcal{C}_{\beta}}:{{x \in W_{\beta}},{\beta < {({\chi{(X)}})}^{+}}}\}}\). By our construction \({\mathcal{C}_{\alpha} \cap \mathcal{C}_{\beta}} = \varnothing\) for distinct \({\alpha,\beta} < {({\chi{(X)}})}^{+}\). Hence \({|{B{(x_{0})}}|} = {({\chi{(X)}})}^{+}\) for some \(x_{0} \in {\bigcup_{\beta < {({\chi{(X)}})}^{+}}\mathcal{C}_{\beta}}\). A contradiction.

We can prove this theorem by the slight modification of the proof of Theorem 2.12 presented in this section to obtain the contradiction with \({\psi{(X)}} \leqslant {H\psi{(X)}} \leqslant {\chi{(X)}}\).

By the result from [11] we have \({wL{(X)}} \leqslant {c{(X)}}\) for each topological space. By Theorem 2.10 we have \({wL{(X_{\delta})}} \leqslant {c{(X_{\delta})}} \leqslant 2^{c{(X)}}\).