Examples:: *[[Examples]]*
Nonexamples:: *[[Nonexamples]]*
Constructions:: *[[Constructions|Used in the construction of...]]*
Specializations:: *[[Specializations]]*
Generalizations:: [[Euclidean diffeomorphism]]
Justifications and Intuition:: *[[Justifications and Intuition]]*
Properties:: *[[Properties]]*
Sufficiencies:: [[Euclidean diffeomorphism|diffeomorphisms are homeomorphisms]]
Equivalences:: *[[Equivalences]]*
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> [!definition] Definition. ([[homeomorphism]])
> Let $X$ and $Y$ be [[topological space|topological spaces]]. A map $f: X \to Y$ is called a **homeomorphism** provided that
> - $f$ is [[bijection|bijective]];
> - $f$ is [[continuous]];
> - $f^{-1}$ is [[continuous]].
>
>
> The postfix 'morphism' indicates the **homeomorphisms** preserve the *topological structure* of the spaces involved (compare to [[group homomorphism]], [[Euclidean diffeomorphism|diffeomorphism]]).
>
The **homeomorpism [[group]]** of a [[topological space]] $X$ is its [[automorphism|automorphism group]] as an object in the [[category]] $\mathsf{Top}$. Explicitly, it is sometimes denoted $\text{homeo}(X)$, and is the set of [[homeomorphism|homeomorphisms]] from $X$ to itself endowed with function composition as the [[binary operation|group operation]].
^17e02c
> [!equivalence]
> Equivalently, $f:X \to Y$ is a **homeomorphism** provided that it is a bijection for which $f(U)$ is [[open set|open]] *iff* $U$ is [[open set|open]].
> > [!intuition]
> So, **homeomorphisms** give us bijective correspondences not only between $X$ and $Y$ but between the collections of *open sets* of $X$ and $Y$. As such, any 'topological property of $X