In a topological space \(X\), a subset \(A\subseteq X\) is called clopen if it is both open and closed. The adjective is a contraction of two familiar words: closed and open.
There is no contradiction here. A set being closed does not mean it cannot be open: the complement of an open set is closed, and conversely. In particular, \(\varnothing\) and \(X\) are always clopen.
A connectedness connection
A space \(X\) is connected precisely when its only clopen subsets are \(\varnothing\) and \(X\). The nontrivial clopen subsets of a disconnected space give a separation into two disjoint nonempty open pieces.
For example, in the usual topology on \(\mathbb R\), only the empty set and the whole real line are clopen. In a discrete space, however, every subset is clopen.
Why call a website Clopenly?
The name is a small piece of mathematical wordplay: clopen, with an adverbial ending that also echoes openly. It reflects the hope that the site can be a place for sharing mathematics openly, whether the thoughts are polished theorems or unfinished questions.