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Let $X$ be a topological space. Then

trivial topology $T$ = $\{\phi,X\}$

discrete topology $T$ = family of all subsets of $X$

standard topology $T$ = ??collection of all open intervals of X??

I understand the trivial and discrete topology but I don't know how to approach to the standard topology. Can someone give me a simple example of standard topology?

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  • $\begingroup$ The trivial topology is also called the indiscrete topology. FYI. $\endgroup$ – Henno Brandsma Apr 28 '18 at 23:10
  • $\begingroup$ The trivial topology is also called the anti-discrete, or coarse topology. $\endgroup$ – DanielWainfleet Apr 29 '18 at 4:30
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There is no such thing as the standard topology on any set $X$. If $X=\mathbb R$, then the standard topology is the topology whose open sets are the unions of open intervals. More generaly, if $X\subset\mathbb R$, then the standard topology is the topology whose open sets are the unions of sets of the type $(a,b)\cap A$, with $a,b\in\mathbb R$ and $a<b$.

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  • $\begingroup$ Ok. Thank you. Maybe my professor used the term in different meaning. The definition I wrote above is what I found online. $\endgroup$ – Anthony Apr 28 '18 at 20:10
  • $\begingroup$ If my answer was useful, perhaps that you could mark it as the accepted one. $\endgroup$ – José Carlos Santos Apr 28 '18 at 20:11
  • $\begingroup$ How do I do that? $\endgroup$ – Anthony Apr 28 '18 at 20:12
  • $\begingroup$ Oh I found it thank you $\endgroup$ – Anthony Apr 28 '18 at 20:13

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