Weight and character of the Kojman-Shelah scale subspace

Fix $x\in X$ and a finite $m$ with $\omega <\mathrm{cf}(x(n))<{\mathrm{\aleph }}_m$ for every $n\in B$, as in F1. Necessarily $m\ge 2$. Partition $B$ into $B_i=\{n:\mathrm{cf}(x(n))={\mathrm{\aleph }}_i\}$, $1\le i<m$, using F4. For every $n\in B_i$ choose by F4 and A1 a strictly increasing cofinal map $e_n:{\mathrm{\aleph }}_i⟶x(n)$. For each tuple $\gamma =({\gamma }_i)$ with ${\gamma }_i<{\mathrm{\aleph }}_i$ at each nonempty stratum define $b_{\gamma }(n)=e_n({\gamma }_i)$ for $n\in B_i$. Then $b_{\gamma }<x$, and $V_{\gamma }=(b_{\gamma },x]\cap X$ is an open neighborhood of $x$ by F3. Given any $g<x$, for each $n\in B_i$ take the least ${\eta }_n<{\mathrm{\aleph }}_i$ with $g(n)<e_n({\eta }_n)$; this exists because $x(n)$ is a limit and $e_n$ is cofinal. Countability of $B_i$ and uncountable regularity F5 give ${\gamma }_i={\sup }_{n\in B_i}{\eta }_n<{\mathrm{\aleph }}_i$ by F4. Monotonicity yields $b_{\gamma }(n)\ge e_n({\eta }_n)>g(n)$, so $V_{\gamma }\subseteq (g,x]\cap X$. F3 shows the $V_{\gamma }$ form an actual local base. Empty strata contribute no tuple coordinate.

By F2 choose $x_0\in X$, for example a point above the zero product function. For $n\in B$ let $x_n$ agree with $x_0$ except that $x_n(n)={\mathrm{\aleph }}_n$. F5 gives cofinality ${\mathrm{\aleph }}_n$ at that coordinate, and a finite aleph bound greater than both this cardinal and the old uniform bound makes $x_n$ a Rudin point. Hence F1 gives $x_n\in X$. Suppose a neighborhood base $(U_j{)}_{j\in J}$ at $x_n$ had size $\rho <{\mathrm{\aleph }}_n$. It is nonempty, since the neighborhood $X$ must contain a base member. By F3 and A1 choose $g_j<x_n$ with $(g_j,x_n]\cap X\subseteq U_j$. F4–F5 give $\delta ={\sup }_{j\in J}{g}_j(n)<{\mathrm{\aleph }}_n$. Put $\zeta =\delta +{\omega }_1$. F10 and F9 show $\mathrm{\mid }\zeta \mathrm{\mid }\le \max (\mathrm{\mid }\delta \mathrm{\mid },{\mathrm{\aleph }}_1)<{\mathrm{\aleph }}_n$, since $n\ge 2$; therefore $\zeta <{\mathrm{\aleph }}_n$. The final ${\omega }_1$ block is cofinal, and each smaller set is bounded in it by F4–F5, so $\mathrm{cf}(\zeta )={\omega }_1$. Also $\zeta >\delta$, including when $\delta =0$. Let $y$ agree with $x_n$ except for $y(n)=\zeta$. The new cofinality is ${\omega }_1$, so it is an admissible finite modification and belongs to $X$ by F1. It lies in every chosen box: at $n$ its value exceeds $\delta \ge g_j(n)$ and is below the top, and elsewhere it equals $x_n$. Thus $y\in U_j$ for every $j$. But the open box at $x_n$ with lower value $\zeta$ at coordinate $n$ and zero elsewhere excludes $y$. No $U_j$ is contained in it, contradicting the base property. We conclude $\chi (x_n,X)\ge {\mathrm{\aleph }}_n$.

Let $D\subseteq X$ have size less than $\lambda$. Each $d\in D$ has a unique index $\alpha (d)$ with $d{=}^{\ast }{f}_{\alpha (d)}$: existence follows from F1, and two different indices would force eventual strict self-inequality on infinite $B$. By F4–F5 there is $\beta <\lambda$ above all those indices; use $\beta =0$ if $D$ is empty. Consequently every $d\in D$ satisfies $d{<}^{\ast }{f}_{\beta }$. By F2 choose $y\in X$ with $f_{\beta }<y$ pointwise. F3 makes $O=(f_{\beta },y]\cap X$ open, and it is nonempty since $y\in O$. It misses $D$: if $d\in O$, it would be strictly above $f_{\beta }$ everywhere and strictly below it at all but finitely many coordinates, impossible on the infinite $B$. Thus no set of size below $\lambda$ is dense. The whole $X$ is dense in itself and has size $\lambda$ by F2, so $d(X)=\lambda$ by F6 and F8.

The tuple family of step 1.1 has cardinality at most the finite product of the ${\mathrm{\aleph }}_i$ for nonempty strata. At least one stratum is nonempty, since $B$ is infinite. F9 bounds this product by ${\mathrm{\aleph }}_{m-1}<\mu$. By F6–F8, $\chi (x,X)<\mu$ for every $x$, whence $\chi (X)\le \mu$. Choose these local bases for all points, using A1 on the proved nonempty sets of suitable choices. Their union is a global basis: for any open $O$ and $x\in O$, a member of the base at $x$ is contained in $O$. Its size is at most $\mathrm{\mid }X\mathrm{\mid }\cdot \mu =\lambda \cdot \mu =\lambda$ by F2 and F9. Therefore $w(X)\le \lambda$.

The infinite $B\subseteq \omega$ is unbounded; hence ${\sup }_{n\in B}{\mathrm{\aleph }}_n=\mu$. Step 1.2 and the raw-supremum convention F6 give $\chi (X)\ge \mu$. Combined with step 2.1, this proves $\chi (X)=\mu$, although each local character is strictly smaller. The lower bound uses the specially constructed top-coordinate points; it does not assert a coordinate-cofinality lower bound at an arbitrary point.

If a global basis had size less than $\lambda$, choose one point from each nonempty member using A1. The resulting set has size less than $\lambda$ and is dense: each nonempty open set contains a nonempty basis member and hence its chosen point. This contradicts step 1.3. Therefore $w(X)\ge \lambda$ by F6–F8. Together with step 2.1 this proves $w(X)=\lambda$, while step 3.1 gives the asserted character and step 1.3 gives the density. QED.