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The grid on $\mathbb{R}^2$ is defined by the set of points such that at most one coordinate is not an integer. With this in mind, e endow $\mathbb{R}^\mathbb{N}$ with the product topology, where $\mat …

If $(P,\leq)$ is a poset and $x\in X$, we let $\downarrow x = \{p\in P: p \leq x\}$, and $\uparrow x$ is defined dually. The collection $$\Big\{P\setminus (\downarrow x): x\in P\Big\} \cup \Big\{P\set …

On any poset $(P, \leq)$ we can consider two different topologies that arise directly from the ordering relation. 1) Order convergence topolog $\tau_o(P)$ : By a set filter $\mathcal{F}$ on $P$ we mea …

Let $\newcommand{\N}{\mathbb{N}}\newcommand{\B}{\mathbf{B}}\B(\N)$ be the collection of all bounded functions $f:\N\to\N$. (A function $f:\N\to\N$ is bounded if there is $M\in\N$ such that $f(k) < M$ …

If $X, Y$ are topological spaces, let $\newcommand{\Cont}{\text{Cont}}\Cont(X,Y)$ denote the collection of continous maps $f:X\to Y$, and we endow $\Cont(X,Y)$ with the product topology inherited from …

Are there non-isomorphic $T_0$-spaces $(X_i, \tau_i)$ for $i = 1,2$ such that $\tau_1 \cong \tau_2$ when considered as partially ordered sets?

Let $(X,\tau)$ be a topological space. A retraction is a continuous map $r:X\to X$ such that $r$ is the identity on $\text{im}(r)$. We call $S\subseteq X$ a retract of $X$ if there is a retraction $r: …

Consider the Sierpinski space $\text{S} = (\{0,1\}, \tau)$ where $\tau = \big\{\emptyset,\{0\}, \{0,1\}\big\}$. Endow $\text{S}^\omega$ with the product topology. If $X, Y$ are topological spaces with …

A hypergraph $H =(V, E)$ consists of a set $V$ and a set $E \subseteq {\cal P}(V)$ of subsets of $V$. A hypergraph coloring is a map $c: V\to \kappa$, where $\kappa \neq \emptyset$ is a cardinal and t …

If $X\neq \emptyset$ is a set, we say ${\cal C}\subseteq {\cal P}(X)\setminus \{\emptyset\}$ is a cover if $\bigcup{\cal C} = X$, and ${\cal C}$ is said to be minimal if for all $D\in {\cal C}$ , the …

What is an example of a topological base ${\cal B}$ for $\mathbb{R}$ with the Euclidean topology such that for every $B_1\neq B_2 \in {\cal B}$ we have $B_1\not\subseteq B_2$?

If $X, Y$ are topological, and $R\subseteq X\times Y$ we say that $R$ is continuous (from $X$ to $Y$) if for every $V\subseteq Y$ with $V$ open, we have $$R^{-1}(V) = \{u\in U: \exists v\in V:(u,v)\i …

A topological space $(X,\tau)$ is said to be homogeneous if for any $x,y\in X$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x) = y$. Moreover, we say that $(X,\tau)$ is shrinkable if t …

What are examples of non-homeomorphic connected $T_2$-spaces $(X_i,\tau_i)$ for $i=1,2$ such that the posets $(\tau_1, \subseteq)$ and $(\tau_2,\subseteq)$ are order-isomorphic?

Is there a connected $T_2$-space $(X,\tau)$ with $|X|>1$ and the following property? Whenever $A$ is a subset of $X$ with $|A|<|X|$ and $f:A\to A$ is a bijection, there is a homeomorphism $\varphi:X\ …