= Solution
A <topological space> is <connected> if it is not the union of two disjoint nonempty <open sets>, or equivalently has no nontrivial subset that is both an <open set> and a <closed set>.
Let $K_x$ be the union of all connected subspaces containing $x$. A union of <connected subsets> with a common point is connected: if a <separation of a topological space> existed, every member containing $x$ would lie wholly in the same side. Hence $K_x$ is connected and contains every connected subspace through $x$, so it is the <connected component> of $x$.
If $K_x\cap K_y\ne\varnothing$, their union is connected and maximality gives $K_x=K_y$. Otherwise they are disjoint. Since every $x$ belongs to $K_x$, the connected components \b[partition $X$].
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