I would like to ask a question about clopen (a subset is both open and closed) subsets of a topological space. Let $(X,\tau)$ be a topological space. Denote with $\tau_{clop}$ the all clopen subsets of the space. If only the open subsets of the space is clopen then $\tau=\tau_{clop}$. Is there any specific name of this type of topological spaces?
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If you assume $T_1$ property then the only such spaces are discrete spaces. – geetha290krm Jun 06 '22 at 11:30
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Yes indeed, I just wondered the general specific name of this type of spaces without assuming it is $T_{1]$. Thank you. @geetha290krm – Woodx Jun 06 '22 at 11:33
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These are called "partition spaces", since (equivalently) they have a base, which is a partition of the space, or equivalently, they are Alexandroff (i.e. each point is contained in a minimal open set) and T3. – Ulli Jun 06 '22 at 18:57
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Thank you so much, I will see it. @Ulli – Woodx Jun 07 '22 at 07:08