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We normally define the concept of topological space $(X,\tau)$ before the notion of base for $\tau$. My question is : Let $(X,\tau)$ a topological space Does always exists a base? for $\tau$ different from trivial bases as $\tau$ itself or
$\tau$ $\setminus${$X$} if $\tau$ is $T_1$. Avoiding also trivial cases in which $\tau$ is the indiscrete topology or discrete topology. Given $\mathcal B$ the set of bases for $\tau$ does it have minimal elements?

Asaf Karagila
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giorgiokyn23
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    Does the discrete topology fall under your notion for "trivial"? – Mr.Gandalf Sauron Jan 11 '23 at 12:23
  • Thanks for the observation – giorgiokyn23 Jan 11 '23 at 12:25
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    Can you please edit to clarify your question? "for τ different from..." is not clear. Which topologies do you exclude, and for the remaining ones, which basis? Also, only 1 question please, and present your attempts to solve it. (I think you can easily find a topological space for which the set of bases has no minimal element, so you can discard your last sentence.) – Anne Bauval Jan 11 '23 at 12:51
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    This answers your last sentence: https://math.stackexchange.com/questions/2255981/conditions-on-a-topological-space-implying-that-it-has-a-minimal-basis?rq=1 – Anne Bauval Jan 11 '23 at 12:57
  • Yes this answer, even very well. Thanks – giorgiokyn23 Jan 11 '23 at 13:02

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