A funcoid related to directed topological spaces

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a set $X$, let $\mathcal F(X)$ be all filters on $\mathcal P(X)$, including the improper filter $\bot_X=\mathcal P(X)$. A filter is a nonempty family closed under finite intersections and supersets. Order filters by $\mathcal A\sqsubseteq\mathcal B$ when $\mathcal A\supseteq\mathcal B$ as families. Thus $\bigsqcup$ is intersection of filter families and $\bigsqcap$ is the filter generated by their union; the empty join is $\bot_X$ and the empty meet is $\top_X=\{X\}$. For $E\subseteq X$, $\uparrow E=\{U\subseteq X:E\subseteq U\}$ is its principal filter. Write $\mathcal A\mathrel{\#}\mathcal B$ when every $U\in\mathcal A$ and $V\in\mathcal B$ have $U\cap V\ne\varnothing$, equivalently $\mathcal A\sqcap\mathcal B\ne\bot_X$.

A funcoid $f:X\to Y$ is a pair of maps $\alpha_f:\mathcal F(X)\to\mathcal F(Y)$ and $\beta_f:\mathcal F(Y)\to\mathcal F(X)$ satisfying \[ \mathcal B\mathrel{\#}\alpha_f(\mathcal A) \quad\Longleftrightarrow\quad \mathcal A\mathrel{\#}\beta_f(\mathcal B) \qquad(\mathcal A\in\mathcal F(X),\ \mathcal B\in\mathcal F(Y)). \] Its reverse interchanges $\alpha_f,\beta_f$. Composition satisfies $\alpha_{g\circ f}=\alpha_g\circ\alpha_f$ and $\beta_{g\circ f}=\beta_f\circ\beta_g$. The identity has both maps equal to the identity. The order on $\mathsf{FCD}(X,Y)$ is pointwise order on $\alpha_f$; its meets and joins mean greatest lower bounds and least upper bounds within this complete lattice. A relation $R\subseteq X\times Y$ induces the principal funcoid $\widehat R$ with $\alpha_{\widehat R}(\mathcal A)$ the filter generated by $\{R[U]:U\in\mathcal A\}$ and the analogous inverse-image formula for $\beta$. Here $R[U]=\{y:\exists x\in U,(x,y)\in R\}$.Let $X=\mathbb R\cup\{-\infty,+\infty\}$ with its usual order topology. For each $x\in X$ let $\mathcal N_x$ be its neighborhood filter. Let $R$ be the funcoid specified on principal filters by \[ \alpha_R(\uparrow A)=\bigsqcup_{x\in A}\mathcal N_x\qquad(A\subseteq X). \] Equivalently $R$ is the complete funcoid of this topology. A funcoid $h$ is complete if for every $A\subseteq X$, $\alpha_h(\uparrow A)=\bigsqcup_{x\in A}\alpha_h(\uparrow\{x\})$. Let $Q=\{(x,y)\in X^2:x\ge y\}$ and let $\widehat Q$ be its principal funcoid.

Statement recorded in the supplied source.

Is the lattice meet $R\sqcap^{\mathsf{FCD}}\widehat Q$ complete? This includes $x=-\infty$ and $x=+\infty$; no endpoint is removed. [S2][S3]

Short English statement

Does intersecting the complete funcoid of the extended real line with its order-relation funcoid preserve completeness?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3442 (OPG-60019). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, A funcoid related to directed topological spaces, OPG-60019. Original problem statement, full mathematical discussion, bibliography and comments. https://www.openproblemgarden.org/op/a_funcoid_related_to_directed_topological_spaces.
• S3 Victor Porton, General Topology as Ordered Semigroup Actions, author manuscript, version of 21 July 2026. Basic definitions; pretopology construction; Complete funcoids. https://github.com/vporton/agt-combined/blob/759e45cced7769cf3a84ce60ed6c30f4ce9ffa8e/chap-funcoids.tex.