Is there a first countable, 0-dimensonal, locally compact, lindelöf, non-compact space in which all non-empty open sets have $\pi$-weight $\mathfrak c$?
It also can be seen here. Thanks for your help.
A $\pi$-base for $X$ is a collection $\mathcal V$ of non-empty open sets in $X$ such that if $U$ is any non-empty open set in $X$, then $V \subset U$ for some $V \in \mathcal V$. $\pi w(X)=\min \{|\mathcal V|: \mathcal V \text{ a } \pi \text{-base for } X\} +\omega$.
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I would recommend using Spacebook for any such queries you might have.
EDIT: These are spaces which satisfy all aforementioned properties except for the $\pi$-weight property. Spacebook does not have that as an option.
Yes, there are such spaces. Here is a way (perhaps not the simplest one) of constructing a space with the properties that you want:
1) Let $Z$ be the Alexandroff duplicate of the Cantor space. All that matters here is that $Z$ is compact, zero-dimensional, first-countable and $d(Z)=\mathfrak{c}$, where $d(\cdot)$ stands for density (i.e. least size of a dense subspace).
2) Let $Y=Z^\omega$ with the product topology. Then $Y$ is still compact, zero-dimensional, first-countable and, by a result of D.B. Motorov, $Y$ is also homogeneous (for this we also use that $Z$ has a dense set of isolated points). Also, not only it is true that $d(Y)=\mathfrak{c}$ (since density can only decrease under continuous maps), but in fact $d(U)=\mathfrak{c}$ for any open $U\subseteq Y$ (here we use compactness and homogeneity of $Y$).
3) Fix $p \in Y$ and let $X=Y \setminus \{p\}$.
We now check that $X$ has all the properties we want. Being a subspace of $Y$, $X$ inherits first-countable and zero-dimensional. Being an open subspace, $X$ also inherits locally compact and "$d(U)=\mathfrak{c}$ for any open $U\subseteq X$". Since $Y$ is compact homogeneous, $p$ is not isolated in $Y$ and therefore $X$ is not compact; however $X$ is Lindelof, being a countable union of compact sets. Finally note that in a first countable space, the $\pi$-weight and the density coincide.
The fact that the $\omega$-th power of a compact zero-dimensional first-countable space with a dense set of isolated points is homogeneous was announced (without proof) by Motorov in "Zero-dimensional and linearly ordered bicompacta: properties of homogeneity type". This was improved latter by Dow and Pearl in "Homogeneity in powers of zero-dimensional first-countable spaces" where they show that the $\omega$-th power of any zero-dimensional first-countable space is homogeneous.