Blog

I would like a good estimate for the number $ f(n)$ of functions $ \varphi\colon [n]\to 2^{[n]}$ , where $ [n]=\{1,2,\dots,n\}$ and $ 2^{[n]}$ is the set of all subsets of $ [n]$ , satisfying for all $ i\in [n]$ the two conditions $ $ \varphi(i) \subseteq \{i+1,i+2,\dots,n\} $ $ $ $ j\in \varphi(i) \RightarrowRead more

Given a poset $ (P,\leq)$ the interval topology $ \tau_i(P)$ on $ P$ is generated by $ $ \{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$ $ where $ \downarrow x = \{y\in P: y\leq x\}$ and $ \uparrow x = \{y\in P: y\geq x\}$ . Let $ f: P\to Q$ beRead more