69 problems

de Groot's conjecture. Every $n$-dimensional absolute suspension is homeomorphic to the $n$-sphere.

Nagata's conjecture. Every $M$-space is homeomorphic to a closed subspace of the product of a countably compact space and a metric space.

Let $X$ be a metrizable space, and let $F_4(X)$ denote the subspace of the free topological group $F(X)$ consisting of words of reduced length at most $4$ with respect to the free…

Let $d,s\geq 1$. Suppose that $L_1,\ldots,L_d$ are linearly ordered compact spaces and there is a continuous surjection … where $K_1,\ldots,K_{d+s}$ are infinite compact spaces. Ma…

One-point connectification conjecture. Let $i=3$. A locally connected $T_i$-space has a $T_i$ one-point connectification if and only if it has no compact (or almost compact) compon…

Scheepers Conjecture. A topological space is a wQN-space if and only if it satisfies $S_1(\Gamma,\Gamma)$.

Let $K$ be a nonempty Valdivia compact space satisfying the countable chain condition (ccc). Valdivia compacta ccc dichotomy conjecture. Either $K$ has a $G_\delta$ point or $K$ ad…

Let $\mathbf{CH}$ be the category of compact Hausdorff spaces and continuous maps. A category is Stone-like when its abstract ultracoproduct construction satisfies the relevant Sto…

A first countable (Tychonov) space is a topological space in which every point has a countable local base. A point-countable $\pi$-base is a $\pi$-base such that each point belongs…

Let $G$ be a separable, metrizable, connected topological group of finite dimension, and suppose that $G$ is complete. An arc component of the identity is the subset of points that…

Let $X$ be a compact topological space, and suppose every closed subset of $X$ is a $G_\delta$ set, meaning a countable intersection of open subsets of $X$. Fremlin's two-to-one co…

Let $X$ be a regular Hausdorff space. A space is hereditarily separable if every subspace is separable, and hereditarily Lindelöf if every subspace is Lindelöf. Hajnal–Juhász basis…

Let $X$ be a compact homogeneous space of countable tightness, meaning that $t(X)=\aleph_0$: whenever $A\subseteq X$ and $x\in\overline{A}$, there is a countable $B\subseteq A$ suc…

Compactness–sequential compactness conjecture. For first countable perfectly normal spaces, compactness and sequential compactness should coincide.

Simon's conjecture. If there is no sequence in $\mathbb{N}$ converging to a point of $b\mathbb{N}\setminus\mathbb{N}$, then there is a continuous map

Watson's conjecture. Such an example exists in ${\sf ZFC}$, probably even a completely regular one, and its existence depends on hard finite combinatorics.

Dow's conjecture. There is a model of set theory satisfying that if $X$ is compact and $\omega \times X$ has remote points, then $X$ has an open subset with countable cellularity.

A space is locally fine when its uniformity is locally fine, and a space is subfine when it is uniformly homeomorphic to a subspace of a fine space. Isbell's conjecture. Every loca…

Hereditarily indecomposable-space conjecture. Hereditarily indecomposable spaces also satisfy the IVT for monic polynomials.

Let $A$ be the subset appearing in the FMQ construction, let $\mathcal{D}_Y$ be its specified family of data, and set … Here, $k_2$-Hausdorff means that the kernel of every continu…

A topological space is Baire if every countable intersection of open dense subsets is dense. An $\mathcal{N}$-space is the class of spaces defined in the source. Christensen's conj…

Non-determinacy problem. The game $G_{\omega}(Y,\mathbb{A})$ is not determined.

A topological space is crowded if it has no isolated points, pseudocompact if every real-valued continuous function on it is bounded, and ω-resolvable if it contains countably many…

Equivalence conjecture for $\sigma E$.

Characterization problem. Find a characterization of trees which are $Q$-spaces or $\Delta$-spaces.