= Homeomorphism
{title2=$f:X\xrightarrow{\sim}Y$}
{wiki}
= Homeomorphic
{synonym}
A homeomorphism is a <bijection> between <topological spaces> such that both $f$ and $f^{-1}$ are <continuous maps>. Equivalently, for every <subset> $U\subseteq X$, $U$ is an <open set> if and only if $f(U)$ is an <open set> in $Y$. A <homeomorphism> therefore transports the entire <topology>, including <connectedness> and <compactness>. It need not preserve numerical <metric> values, and a <bounded set> need not map to a <bounded set>.
For example, an explicit <homeomorphism> between an <open interval> and the <real line> is
$$
f:(0,1)\to\mathbb R,\qquad f(x)=\tan\bigl(\pi(x-1/2)\bigr),\qquad
f^{-1}(y)=\frac12+\frac{\arctan y}{\pi}.
$$
The <tangent function> on $(-\pi/2,\pi/2)$ is strictly increasing and takes every real value; the displayed <arctangent> inverse verifies the <bijection>. Both formulas define <continuous maps> on their stated domains. Computed values are $f(1/4)=-1$, $f(1/2)=0$, and $f(3/4)=1$. Points approach infinity as $x$ approaches an excluded endpoint. Thus a <bounded set> can have the same <topology> as an unbounded one; neither of these <topological spaces> is <compact>.
A useful shortcut is the <compact-to-Hausdorff continuous bijection theorem>: a <continuous> <bijection> from a <compact space> to a <Hausdorff space> is a <homeomorphism>. Its images of <closed sets> are <compact sets>, hence <closed sets> in the target, which proves <continuity> of the inverse. The <indiscrete topology> example shows why the <Hausdorff> assumption matters. https://www.maths.ox.ac.uk/system/files/attachments/Preliminary%20Reading%20%281%29.pdf[Richard Haydon, Oxford topology preliminaries, Theorem 0.3]{ref}
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