= Connected space
{wiki}
= Connected
{synonym}
= Connectedness
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A <topological space> $X$ is <connected> when it has no <separation of a topological space>: no <partition of a set> into two nonempty <open subsets> $U,V$ satisfying
$$
X=U\cup V,\qquad U\cap V=\varnothing.
$$
Here the <union> covers all of $X$, and the <intersection> is the <empty set>, so no point belongs to both pieces. For a <subset> $X$ of a larger <topological space>, "<open>" means <open> in the <subspace topology>: each piece is the <intersection> of $X$ with an ambient <open set>. Both pieces must be <open>; merely dividing a drawing into two colours does not establish a <separation of a topological space>. https://math.mit.edu/~hrm/palestine/munkres-topology.pdf[Munkres, Topology: A First Course, section 3-1, definition on p. 147]{ref}
For example, the <closed interval> $I=[0,1]$ with its usual <subspace topology> from the <real line> is <connected>. The proposed split $[0,1/2)\cup[1/2,1]$ fails: the second <subset> is not <open> in $I$, since every <neighbourhood> of $1/2$ in $I$ contains points less than $1/2$. More generally, a <connected subset of the real line> is precisely a <real interval>. In contrast, for the <union>
$$
D=[-2,-1]\cup[1,2],\qquad U=D\cap(-3,0)=[-2,-1],\qquad V=D\cap(0,3)=[1,2],
$$
the <open intervals> $(-3,0)$ and $(0,3)$ exhibit $U,V$ as nonempty <open subsets> in the <subspace topology> of $D$. Their <intersection> is empty and their <union> is $D$, so $D$ is not <connected>. Although $U,V$ are <closed intervals> in the <real line>, they are also <open> within $D$.
\Image[/connected-and-disconnected-sets.png]
{title=A <connected> <closed interval> and a <union> of two <closed intervals> admitting a <separation of a topological space> in the <subspace topology>}
Why is <connectedness> useful? A <continuous image of a connected space> is <connected>: a proposed <separation of a topological space> in the <image> would pull back through the <continuous map> to a forbidden <separation of a topological space> in its source. Thus a <continuous map> from a <real interval> to the <real line> cannot skip values between two attained values. This is the <intermediate value theorem>. For a computed example, the <continuous function> $f(x)=x^2$ on $[0,1]$ has <image> $[0,1]$; the value $1/2$ occurs at $x=1/\sqrt2\approx0.707107$. <Connectedness> guarantees an input attaining that value even when we have no formula for it.
<Path connectedness> is a stronger condition: it supplies a <continuous path> between every pair of points. The <topologist's sine curve> below shows that a <connected space> need not be a <path-connected space>.
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