Timeline for Which topologies admit exactly one open singleton and have more than half of the possible subsets open?

6 events

when toggle format	what		by	license	comment
Jun 17 at 23:27	answer	added	Robin Saunders		timeline score: 1
Jun 17 at 19:12	comment	added	Paul Sinclair		Thus you can start with the topologies of $Y$ with no discrete points and more than $2^{n-2}$ open sets, then determine how many ways they can be completed to form a desired topology on $X$.
Jun 17 at 19:11	comment	added	Paul Sinclair		Let $Y = X\setminus \{a_0\}$. For each of your topologies $\tau$, the subspace topology $\tau_Y$ on $Y$ has no single point open sets. For a given $\tau_Y$, all the topologies on $\tau$ that induce it can be constructed by first including the unions of each set in $\tau_Y$ with $\{a_0\}$, Then choose which sets in $\tau_Y$ you want to also be in $\tau$ directly, with the two provisios that $\emptyset$ must be included, and if $U \in \tau_Y$ is in $\tau$ then so must be every open subset of $U$ in $Y$.
Jun 16 at 20:14	comment	added	Ulli		This paper characterizes all topologies on an n-element set with $2^{n-1} < \|\tau\|$, at least implicitly.
Jun 16 at 8:43	history	edited	Martin Sleziak	CC BY-SA 4.0	capitalization
Jun 16 at 8:42	history	asked	amir bahadory	CC BY-SA 4.0