= Path-connected space
{wiki}
= Path-connected
{synonym}
= Path connectedness
{synonym}
A <topological space> $X$ is <path-connected> if, for every $a,b\in X$, there is a <continuous map>
$$
\gamma:[0,1]\longrightarrow X,\qquad\gamma(0)=a,\qquad\gamma(1)=b.
$$
This <continuous path> must stay within $X$ for its entire duration. Its parameter need not represent physical time, and <continuity> does not require a <derivative> or finite <arc length>. https://math.mit.edu/~hrm/palestine/munkres-topology.pdf[Munkres, Topology: A First Course, section 3-2, definition on p. 155]{ref}
In the <closed interval> $[0,1]$, any $a,b$ are joined by $\gamma(t)=(1-t)a+tb$. For $a=1/4$ and $b=3/4$, this becomes $\gamma(t)=1/4+t/2$, so $\gamma(0)=1/4$, $\gamma(1/2)=1/2$, and $\gamma(1)=3/4$. All its values stay in $[0,1]$. In the <union> $D=[-2,-1]\cup[1,2]$, however, a <continuous path> from $-1$ to $1$ would attain $0$ by the <intermediate value theorem>, although $0\notin D$. Thus $D$ is not <path-connected>.
Every <path-connected space> is a <connected space>. Proof outline: if $X=U\cup V$ admitted a <separation of a topological space>, choose $a\in U$ and $b\in V$. A <continuous path> between them would have nonempty <open> <inverse images> $\gamma^{-1}(U),\gamma^{-1}(V)$ with empty <intersection> and <union> $[0,1]$. This contradicts the <connectedness> of that <closed interval>.
The converse fails: the <topologist's sine curve> is <connected> but has two <path components>. <Path connectedness> is useful when we need an actual <continuous path> of allowed positions, rather than only the absence of a <separation of a topological space>. For instance, when modelling allowed positions as a <topological space>, membership in the same <path component> answers whether one can move continuously between two positions while remaining in that <topological space>.
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