= Path component
{title2=$P_x$}
= Path-connected component
{synonym}
A <path component> of a <topological space> $X$ is a nonempty <subset> maximal among its <path-connected> <subsets>. The <path component> through $x$ can be described as
$$
P_x=\{y\in X:\text{there is a continuous path in }X\text{ from }x\text{ to }y\}.
$$
Every point along such a <continuous path> also belongs to $P_x$: the initial part of the <path> reaches that point. Thus the <paths> witnessing membership stay inside $P_x$. This makes <path components> useful for deciding which allowed positions can be reached from one another by continuous motion.
Every <path component> lies inside one <connected component>, because every <path-connected space> is <connected>. Thus each <connected component> is a <union> of <path components>. In a <locally path-connected space> the two classifications agree, by <connected locally path-connected spaces are path-connected>. For a <subset> of the <real line> with its usual <subspace topology>, the classifications also agree: every <connected component> is a <real interval>, and interpolation supplies a <continuous path> within it.
The <topologist's sine curve> shows why the distinction matters in general. Use specifically the closed version
$$
G=\{(x,\sin(1/x)):0<x\le1\},\qquad
L=\{0\}\times[-1,1],\qquad T=G\cup L,
$$
with the <subspace topology> from the usual <Euclidean plane>. It has one <connected component>, $T$, but two <path components>, $G$ and $L$. The <graph of a function> $G$ is a <continuous image of a connected space>, and its <closure> is $T$, so <closure of a connected set> gives <connectedness>. Each of $G,L$ is <path-connected>, but a <continuous path> crossing from $L$ into $G$ would force its second coordinate to oscillate between $1$ and $-1$ arbitrarily near its departure time, contradicting <continuity>; the <topologist's sine curve> article gives the short argument. Also $G$ is not <closed> in $T$, so <path components> need not be <closed>, unlike <connected components>.<General Topology by Tom Leinster>[p. 78, and the discussion after Lemma C4.12, p. 84: the sine-curve obstruction and nonclosed path components; here the positive graph is restricted to $0<x\le1$]{ref}
Being joined by a <continuous path> is an <equivalence relation>. Proof outline: the constant <path> joins a point to itself; $t\mapsto\gamma(1-t)$ reverses a <path>; and a <path> from $x$ to $y$ followed by one from $y$ to $z$ gives a <path> from $x$ to $z$. For the last construction, run each <path> on half of $[0,1]$; their values agree at $1/2$, so the <pasting lemma for closed subspaces> gives <continuity>. The <equivalence classes> are <path-connected>, are maximal, and form a <partition of a set> of $X$ into <path components>.<General Topology by Tom Leinster>[p. 84: path equivalence and maximal path-connected subsets]{ref}
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