# Topological cardinal function : The grasp

Def.: For a topological space $$S$$ with weight $$w(S)=\kappa$$, I define the grasp $$g(S)$$ to be the least infinite cardinal $$\gamma$$ such that $$S$$ has a base $$\mathscr{B}$$ with $$|\mathscr{B}|=\kappa$$ such that every open set is the union of at most $$\gamma$$ members of $$\mathscr{B}$$.

When I spoke to Franklin Tall (Prof. Emeritus, U.of Toronto) last fall, he said he hadn't heard of it. (His specialty is point-set topology and set-theory).

I can show the following by elementary means:

• [R1] $$g(S) \le w(S)$$ (obviously).

• [R2] If $$E$$ is a subspace of $$D$$ and $$w(E)=w(D)$$, then $$g(E)\le g(D)$$.

• [R3] Let $$D[\mu]$$ be the discrete space of cardinality $$\mu$$, where $$\mu$$ is an infinite cardinal. If $$w(S)=\mu$$, then $$g(S)\le g(D[\mu])$$.

• [R4] If $$\mathscr{T}$$ is the topology on $$S$$ then $$|\mathscr{T}|\le w(S)^{g(S)}$$ (obviously).

• [R5] Let the infinite cardinal $$\kappa$$ have either the discrete topology or the order topology. Then $$\operatorname{cf}(\kappa)\le g(\kappa)\le \kappa$$, and if $$\kappa$$ is a strong limit cardinal then $$g(\kappa)=\operatorname{cf}(\kappa)$$. NOTE: This last part of [R5] is useful to distinguish the grasp from other topological cardinal functions.

Questions:

• [Q1] For a topology $$\mathscr{T}$$ on $$S$$, the grasp $$g(S)$$ is not necessarily the least infinite $$\lambda$$ such that $$|\mathscr{T}|\le w(S)^ \lambda$$, for if $$\mathscr{T}$$ is the discrete topology on $$\omega _1$$ and if $$2^\omega = 2^{\omega _1}$$, then $$|\mathscr{T}|=\omega _1^ \omega$$, but the grasp is $$\omega_1$$, not $$\omega$$. Is there an example like this in $$\mathsf{ZFC}$$?

• [Q2] What are the possible consistent values for the grasp of the discrete space $$D[2^ \omega]$$ aside from the value $$2^ \omega$$?

Examples:

• [E1] The Sorgenfrey line has weight $$2^ \omega$$ and grasp $$\omega$$. So in [R3] we may have, by [R5], that $$g(S) < \operatorname{cf}(\chi)\le \chi=g(D[w(S)])$$.

• [E2] By [R1] and [R2],the Nimitzky plane $$P$$ has $$g(P)=g(D[2^ \omega])$$.

• If $S$ has a base $B$ such that $\lvert B\rvert\le k,$ then by definition of weight, we have $\lvert B\rvert=k,$ do we not? Commented Aug 17, 2015 at 18:55
• @user254665: This site has a MathJax tutorial. \\ The one obvious comment is that $h(X)\le g(X)$, where $h(X)$ is the height (= hereditary Lindelöf degree). Commented Aug 17, 2015 at 19:53