Unsolved Problems

Question $S(S(f)) = S(f)$ for every endo-reloid $f$?...

Conjecture Does every Jordan curve have 4 points on it which form the vertices of a square?...

Problem Given a link $L$ in $S^3$, let the symmetry group of $L$ be denoted $Sym(L) = \pi_0 Diff(S^3,L)$ ie: isotopy classes of diffeomorphisms of $S^...

Problem Does there exist a subset of $\mathbb R^3$ such that its fundamental group has an element of finite order?...

Problem Ian Agol and Matthias Goerner observed that the 4x5 chessboard complex is the complement of many distinct links in the 3-sphere. Their observa...

Question $( \mathsf{RLD})_{\mathrm{out}} (f \cap^{\mathsf{FCD}} ( \mathcal{A} \times^{\mathsf{FCD}} \mathcal{B})) = (( \mathsf{RLD})_{\mathrm{out}} f)...

Conjecture Let $C$ be a Cantor set embedded in $\mathbb{R}^n$. Is there a self-homeomorphism $f$ of $\mathbb{R}^n$ for every $\epsilon$ greater than $...

Conjecture Does any path-connected, compact set in the plane which does not separate the plane have the fixed point property? A set has the fixed poi...

Conjecture Let $G$ be a locally compact topological group. If $G$ has a continuous faithful group action on an $n$-manifold, then $G$ is a Lie group....

Conjecture $\mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}}_F \mathcal{B} \subset \mathcal{A} \ltimes \mathcal{B} \subset \mathcal{A} \t...

Conjecture (solved) If $a \times^{\mathsf{\ensuremath{\operatorname{RLD}}}} b \subseteq \left( \mathsf{\ensuremath{\operatorname{RLD}}} \right)_{\ensu...

Let $\mho$ be a set, $\mathfrak{F}$ be the set of filters on $\mho$ ordered reverse to set-theoretic inclusion, $\mathfrak{P}$ be the set of principal...

Conjecture The poset of completary multifuncoids of the form $(\mathscr{P}\mho)^n$ is for every sets $\mho$ and $n$: - atomic; - atomistic. See belo...

Conjecture The poset of multifuncoids of the form $(\mathscr{P}\mho)^n$ is for every sets $\mho$ and $n$: - atomic; - atomistic. See below for defin...

Conjecture Let $F$ is a family of multifuncoids such that each $F_i$ is of the form $\lambda j \in N \left( i \right): \mathfrak{F} \left( U_j \right)...

Conjecture Let $f_1$ and $f_2$ are monovalued, entirely defined funcoids with $\operatorname{Src}f_1=\operatorname{Src}f_2=A$. Then there exists a poi...

Conjecture For composable reloids $f$ and $g$ it holds - $\operatorname{Compl} ( g \circ f) = ( \operatorname{Compl} g) \circ f$ if $f$ is a co-compl...

Conjecture Every metamonovalued funcoid is monovalued. The reverse is almost trivial: Every monovalued funcoid is metamonovalued....

Conjecture Every metamonovalued reloid is monovalued....

Let $\delta$ be a proximity. A set $A$ is connected regarding $\delta$ iff $\forall X,Y \in \mathscr{P} A \setminus \{ \emptyset \}: \left( X \cup Y ...

Conjecture Let $f$ is a $T_1$-separable (the same as $T_2$ for symmetric transitive) compact funcoid and $g$ is a uniform space (reflexive, symmetric,...

Definition $\square f = \bigcap^{\mathsf{RLD}} \mathrm{up}^{\Gamma (\operatorname{Src} f; \operatorname{Dst} f)} f$ for reloid $f$. Conjecture $(\mat...

Let $\mathfrak{A}$ be an indexed family of sets. Products are $\prod A$ for $A \in \prod \mathfrak{A}$. Hyperfuncoids are filters $\mathfrak{F} \Gam...

Conjecture $f \sqcap \bigsqcup S = \bigsqcup \langle f \sqcap \rangle^{\ast} S$ for principal funcoid $f$ and a set $S$ of funcoids of appropriate sou...

Conjecture Let $R$ be the complete funcoid corresponding to the usual topology on extended real line $[-\infty,+\infty] = \mathbb{R}\cup\{-\infty,+\in...

Conjecture For every composable funcoids $f$ and $g$ $$(\mathsf{RLD})_{\mathrm{out}}(g\circ f)\sqsupseteq(\mathsf{RLD})_{\mathrm{out}}g\circ(\mathsf{R...

Define for posets with order $\sqsubseteq$: - $\Phi_{\ast} f = \lambda b \in \mathfrak{B}: \bigcup \{ x \in \mathfrak{A} \mid f x \sqsubseteq b \}$; ...

Warning: This formulation is vague (not exact). Question Characterize the set $\{f\in\mathsf{FCD} \mid (\mathsf{RLD})_{\mathrm{in}} f=(\mathsf{RLD})_...