= Quotient topological space
{title2=$X/{\sim}$}
= Quotient space
{synonym}
A <quotient topological space> is a <quotient set> equipped with the <quotient topology>. For a concrete gluing, give the <closed interval> $I=[0,1]$ its usual <subspace topology>, identify $0\sim1$, and leave every interior point in its own <singleton> <equivalence class>. The resulting <quotient space> is a <circle>: the two endpoints become one point, while interior points remain distinct.
\Image[/quotient-interval-endpoints-circle.png]
{title=Gluing the interval endpoints produces a circle, with matching colors showing the projection}
The exact <continuous map> to the <unit circle> in the plane is
$$
p(t)=\bigl(\cos(2\pi t),\sin(2\pi t)\bigr),\qquad
\bar p([t])=p(t).
$$
Here <sine> and <cosine> give $p(0)=p(1)=(1,0)$, and $p(s)=p(t)$ holds exactly when $s=t$ or $\{s,t\}=\{0,1\}$. Thus $p$ induces a <bijection> $\bar p:I/{\sim}\to S^1$. <Topology and Groups/3.01 Quotient topology>{ref} The <universal property of the quotient topology> makes $\bar p$ <continuous>. The <quotient of a compact space> is <compact>, and the <unit circle> is <Hausdorff>, so the <compact-to-Hausdorff continuous bijection theorem> makes $\bar p$ a <homeomorphism>. This verifies the <topology>, not merely the point identifications.
To inspect an <open> <neighbourhood> of the seam, choose $0<\varepsilon<1/2$ and let $U_\varepsilon=p([0,\varepsilon)\cup(1-\varepsilon,1])$. Its full <preimage> is
$$
p^{-1}(U_\varepsilon)=[0,\varepsilon)\cup(1-\varepsilon,1].
$$
Both pieces are <open> in the <subspace topology> on $I$, so $U_\varepsilon$ is <open> in the <circle>. In contrast, $A=[0,\varepsilon)$ is <open> in $I$, but
$$
p^{-1}\bigl(p(A)\bigr)=[0,\varepsilon)\cup\{1\}
$$
is not <open> in $I$: it contains $1$ without any <neighbourhood> of $1$. Hence $p(A)$ is not <open> in the <circle>. A <quotient map> can therefore fail to be an <open map>.
\Image[/quotient-circle-open-preimages.png]
{title=An open neighbourhood crosses both sides of the glued seam; a one-sided arc fails the full-preimage test}
For a <continuous function> $f:I\to\mathbb R$, the only gluing condition is $f(0)=f(1)$. For example, $f(t)=\cos(2\pi t)$ defines a <continuous function> on the <circle>, whereas $f(t)=t$ does not. This lets us use an <interval> as a coordinate for a <periodic function> while making the seam invisible. More general <quotient spaces> encode glued geometric pieces in the same way: describe <continuous maps> before gluing and check that their values agree on every <equivalence class>.
The <Hausdorff> property need not survive a <quotient topology>. In the <non-Hausdorff quotient of the real line by rational translation>, $x\sim y$ means $x-y\in\mathbb Q$. A nonempty <open> <preimage> $W$ must contain all translates by <rational numbers> of any <open interval> inside it. The <density of the rational numbers> then forces $W=\mathbb R$: for any $y$, some <rational number> $r$ has $y-r$ in that <interval>. Thus this <quotient space> has the <indiscrete topology>. The <irrationality of the square root of two> gives distinct <equivalence classes> for $0$ and $\sqrt2$, so this <quotient space> is not <Hausdorff>. Unlike the endpoint-gluing <circle>, this gluing identifies a <dense subset> with each point.
Back to article page
