A [topological space](/page/Topological%20Space) remembers which subsets are open, and therefore remembers which maps are continuous, which sequences or nets converge, which subsets are compact, and which pieces are connected. It does not remember lengths, angles, differentiability, or coordinates. A homeomorphism is the map that preserves exactly this level of structure: it identifies two spaces as the same object from the viewpoint of [Topology](/page/Topology). In analysis this distinction matters constantly. The interval $(0,1)$ and the real line $\mathbb{R}$ have different sizes as metric intervals but the same topology; a function can stretch one onto the other without tearing or gluing. By contrast, the half-open interval $[0,1)$ cannot be topologically reshaped into $(0,1)$, because its endpoint is detected by how point-complements behave: removing that endpoint leaves a connected space, while removing an interior point of $(0,1)$ disconnects it. Homeomorphisms give a precise language for these comparisons, sitting between [Continuity](/page/Continuity), [Open Set](/page/Open%20Set), [Closed Set](/page/Closed%20Set), and the invariants of topological spaces. ## Definition A continuous bijection alone is not enough to express topological sameness: it may identify the underlying sets while giving the target a topology whose open sets are not transported back correctly. The inverse must also respect the topology. This two-sided condition is needed because topological sameness should allow arguments to move from $X$ to $Y$ and then return without losing information. [definition: Homeomorphism] Let $(X, \tau_X)$ and $(Y, \tau_Y)$ be topological spaces. A function $f: X \to Y$ is a homeomorphism from $(X, \tau_X)$ to $(Y, \tau_Y)$ if $f$ is bijective, $f$ is continuous, and the inverse function $f^{-1}: Y \to X$ is continuous. [/definition] The definition treats $f$ and $f^{-1}$ symmetrically. A homeomorphism is therefore not merely a well-behaved map from $X$ to $Y$; it is a reversible translation of the entire open-set structure of $X$ into that of $Y$. To compare spaces rather than individual maps, we need a name for the existence of such a reversible translation. This relation is the analogue, in topology, of isomorphism in algebra or linear isomorphism in linear algebra. [definition: Homeomorphic Spaces] Let $(X, \tau_X)$ and $(Y, \tau_Y)$ be topological spaces. The spaces $X$ and $Y$ are homeomorphic if there exists a homeomorphism $f: X \to Y$. [/definition] When $X$ and $Y$ are homeomorphic, we often write that $X \cong Y$ in a topological context. The symbol must be read from context, since it can also denote other kinds of isomorphism in algebraic settings. Once spaces are classified up to homeomorphism, the next question is which statements survive this classification. This motivates isolating the properties that depend only on topology, rather than on a chosen metric, embedding, or coordinate system. [definition: Topological Property] A property $P$ of topological spaces is a topological property if, whenever $X$ and $Y$ are homeomorphic topological spaces, $X$ has property $P$ if and only if $Y$ has property $P$. [/definition] Compactness, connectedness, [path connectedness](/page/Path%20Connectedness), and the Hausdorff property are topological properties. Boundedness of a subset of a metric space is not a topological property, since boundedness depends on a chosen metric rather than only on the open sets. ## Equivalent Characterisations The definition uses continuity of both directions, but in practice homeomorphisms are often recognised by how they move open sets. This is especially useful when the inverse is not written explicitly. To express that method, we first name maps that send open sets forward to open sets. [definition: Open Map] Let $(X, \tau_X)$ and $(Y, \tau_Y)$ be topological spaces. A function $f: X \to Y$ is an open map if $f(U) \in \tau_Y$ for every $U \in \tau_X$. [/definition] For a bijection, openness of the forward map is exactly the missing continuity condition for the inverse. This gives a practical test for homeomorphism, especially when direct images of basic open sets are easier to control than inverse formulas. [quotetheorem:8835] A parallel recognition method uses closed sets, and that method interacts well with compactness arguments. To use it precisely, we need the corresponding notion of a map that preserves closed sets by direct image. [definition: Closed Map] Let $(X, \tau_X)$ and $(Y, \tau_Y)$ be topological spaces. A function $f: X \to Y$ is a closed map if $f(C)$ is closed in $Y$ for every closed subset $C \subset X$. [/definition] A second practical test for homeomorphism is needed when closed subsets are the available data. The following criterion converts that data into continuity of inverse maps. [quotetheorem:8836] Sometimes neither direct images of all open sets nor direct images of all closed sets is the natural datum. The most intrinsic test asks whether preimages reproduce the target topology exactly. [quotetheorem:8837] This characterisation explains the philosophy of the concept. A homeomorphism is not merely compatible with the two topologies; it transports one topology exactly onto the other. ## Standard Examples The most familiar homeomorphisms in analysis are changes of variables that stretch space without changing its open-set structure. The real line and a bounded open interval provide the model example. [example: Open Interval Homeomorphic to the Real Line] Consider $f:(0,1)\to \mathbb{R}$ defined by \begin{align*} f(x)=\tan(\pi(x-1/2)). \end{align*} For $x\in(0,1)$, we have $-1/2<x-1/2<1/2$, so \begin{align*} -\pi/2<\pi(x-1/2)<\pi/2. \end{align*} The tangent function maps $(-\pi/2,\pi/2)$ bijectively onto $\mathbb{R}$, with inverse $\arctan:\mathbb{R}\to(-\pi/2,\pi/2)$. Thus the proposed inverse is \begin{align*} g(y)=\frac{1}{\pi}\arctan(y)+\frac{1}{2}. \end{align*} Since $-\pi/2<\arctan(y)<\pi/2$, dividing by $\pi$ and adding $1/2$ gives $0<g(y)<1$, so $g(y)\in(0,1)$. For $x\in(0,1)$, \begin{align*} g(f(x))=\frac{1}{\pi}\arctan(\tan(\pi(x-1/2)))+\frac{1}{2}. \end{align*} Because $\pi(x-1/2)\in(-\pi/2,\pi/2)$ and $\arctan$ is the inverse of $\tan$ on that interval, \begin{align*} g(f(x))=\frac{1}{\pi}\pi(x-1/2)+\frac{1}{2}=x. \end{align*} For $y\in\mathbb{R}$, \begin{align*} f(g(y))=\tan\left(\pi\left(\frac{1}{\pi}\arctan(y)+\frac{1}{2}-\frac{1}{2}\right)\right). \end{align*} The expression inside the tangent is $\arctan(y)$, so \begin{align*} f(g(y))=\tan(\arctan(y))=y. \end{align*} Hence $g=f^{-1}$, so $f$ is bijective. The map $f$ is continuous because it is the composition of the affine map $x\mapsto \pi(x-1/2)$ with $\tan$ restricted to $(-\pi/2,\pi/2)$. The inverse $g$ is continuous because it is the composition of $\arctan$ with the affine map $u\mapsto u/\pi+1/2$. Therefore $f$ is a homeomorphism, and $(0,1)$ and $\mathbb{R}$ have the same topology even though one is bounded in its usual metric and the other is not. [/example] This example shows why homeomorphism is not a metric notion. The interval $(0,1)$ is bounded in its usual metric, while $\mathbb{R}$ is not bounded in its usual metric, yet their topological structures agree. Failure examples are just as important, because they reveal which features are genuinely topological. A half-open endpoint cannot be erased by a homeomorphism. [example: Half-Open Interval Not Homeomorphic to an Open Interval] Assume, for contradiction, that there is a homeomorphism $h:[0,1)\to(0,1)$. Put $b=h(0)$, so $b\in(0,1)$. The restriction \begin{align*} h|_{[0,1)\setminus\{0\}}:[0,1)\setminus\{0\}\to(0,1)\setminus\{b\} \end{align*} has inverse $h^{-1}|_{(0,1)\setminus\{b\}}$, and both restricted maps are continuous in the subspace topologies. Hence this restriction is a homeomorphism. Now \begin{align*} [0,1)\setminus\{0\}=(0,1). \end{align*} The interval $(0,1)$ is connected by *Intervals in the Real Line Are Connected*. On the other hand, \begin{align*} (0,1)\setminus\{b\}=(0,b)\cup(b,1). \end{align*} The sets $(0,b)$ and $(b,1)$ are nonempty because $0<b<1$. They are disjoint, and they are open in the subspace $(0,1)\setminus\{b\}$ since \begin{align*} (0,b)=(-1,b)\cap\bigl((0,1)\setminus\{b\}\bigr) \end{align*} and \begin{align*} (b,1)=(b,2)\cap\bigl((0,1)\setminus\{b\}\bigr). \end{align*} Their union is all of $(0,1)\setminus\{b\}$, so they form a separation of that space. Thus $(0,1)\setminus\{b\}$ is disconnected. This contradicts the fact that [homeomorphisms preserve connectedness](/theorems/8840). Therefore $[0,1)$ and $(0,1)$ are not homeomorphic. [/example] The next example shows that a continuous bijection can fail to be a homeomorphism. This is the standard warning against omitting continuity of the inverse from the definition. [example: Continuous Bijection Whose Inverse Is Not Continuous] Let $X=[0,2\pi)$ with the usual [subspace topology](/page/Subspace%20Topology) inherited from $\mathbb{R}$, and let $S^1\subset\mathbb{R}^2$ be the unit circle with the subspace topology. Define $f:X\to S^1$ by \begin{align*} f(t)=(\cos t,\sin t). \end{align*} The coordinate functions $t\mapsto \cos t$ and $t\mapsto \sin t$ are continuous on $\mathbb{R}$, so their restrictions to $[0,2\pi)$ are continuous, and therefore $f$ is continuous as a map into the subspace $S^1$. For every $t\in[0,2\pi)$, \begin{align*} \cos^2 t+\sin^2 t=1. \end{align*} Thus $f(t)\in S^1$. To see that $f$ is injective, suppose $f(t)=f(s)$ with $s,t\in[0,2\pi)$. Then $\cos t=\cos s$ and $\sin t=\sin s$, so \begin{align*} \cos(t-s)=\cos t\cos s+\sin t\sin s=\cos^2 s+\sin^2 s=1. \end{align*} Also, \begin{align*} \sin(t-s)=\sin t\cos s-\cos t\sin s=\sin s\cos s-\cos s\sin s=0. \end{align*} Hence $t-s=2\pi k$ for some integer $k$. Since $s,t\in[0,2\pi)$, we have $t-s\in(-2\pi,2\pi)$, so the only possible multiple of $2\pi$ is $0$. Therefore $t=s$. To see that $f$ is surjective, let $(a,b)\in S^1$. Since $a^2+b^2=1$, there is an angle $\theta\in[0,2\pi)$ with $\cos\theta=a$ and $\sin\theta=b$. Then \begin{align*} f(\theta)=(a,b). \end{align*} So $f$ is bijective. Let $g=f^{-1}:S^1\to[0,2\pi)$. We show that $g$ is not continuous at $(1,0)$. The set \begin{align*} U=[0,1/2)=(-1/2,1/2)\cap[0,2\pi) \end{align*} is open in $[0,2\pi)$, and $g(1,0)=0\in U$. If $g$ were continuous at $(1,0)$, then $g^{-1}(U)$ would contain an open neighborhood of $(1,0)$ in $S^1$. But for every $\varepsilon>0$, choose $\delta$ with $0<\delta<\min\{\varepsilon,1/2\}$. The point \begin{align*} p_\delta=(\cos\delta,-\sin\delta)=f(2\pi-\delta) \end{align*} lies in $S^1$. Its distance from $(1,0)$ is \begin{align*} \|p_\delta-(1,0)\|=\sqrt{(\cos\delta-1)^2+\sin^2\delta}=\sqrt{2-2\cos\delta}=2\sin(\delta/2)\leq \delta<\varepsilon. \end{align*} Thus every neighborhood of $(1,0)$ in $S^1$ contains some $p_\delta$. However, \begin{align*} g(p_\delta)=2\pi-\delta. \end{align*} Since $0<\delta<1/2$, we have $2\pi-\delta>1/2$, so $g(p_\delta)\notin[0,1/2)$. Therefore no neighborhood of $(1,0)$ in $S^1$ is contained in $g^{-1}(U)$, and $g$ is not continuous at $(1,0)$. Thus $f$ is a continuous bijection whose inverse is not continuous, so $f$ is not a homeomorphism. [/example] This example is also a first glimpse of quotient topology. If the endpoints $0$ and $2\pi$ are identified before mapping to $S^1$, the resulting quotient space is homeomorphic to the circle. Some spaces make homeomorphisms abundant because their topologies impose very little structure beyond the underlying set. The [discrete topology](/page/Discrete%20Topology) is the cleanest case. [example: Discrete Spaces and Bijections] Let $X$ and $Y$ be equipped with the discrete topology, so every subset of $X$ is open in $X$ and every subset of $Y$ is open in $Y$. For any function $f:X\to Y$ and any open set $V\subset Y$, the preimage $f^{-1}(V)$ is a subset of $X$. Since the topology on $X$ is discrete, this subset $f^{-1}(V)$ is open in $X$. Thus every function $f:X\to Y$ is continuous. Now suppose $f:X\to Y$ is bijective. Its inverse $f^{-1}:Y\to X$ is a function, and the same argument with $X$ and $Y$ interchanged shows that $f^{-1}$ is continuous: if $U\subset X$ is open, then $(f^{-1})^{-1}(U)$ is a subset of $Y$, hence open in the discrete topology on $Y$. Therefore $f$ is bijective, continuous, and has continuous inverse, so $f$ is a homeomorphism. Conversely, if $f:X\to Y$ is a homeomorphism, then it is bijective by the definition of homeomorphism. Hence, between discrete spaces, the only condition left to check is ordinary bijectivity. [/example] The discrete example shows that homeomorphism can reduce to ordinary bijection when the topology carries no additional restrictions beyond the set structure. In contrast, for familiar subspaces of Euclidean space, the topology carries strong information about local and global shape. ## Properties Homeomorphisms preserve the basic language of topology. This preservation begins with open and closed sets and then propagates to more structured notions such as compactness and connectedness. [quotetheorem:8838] Open and closed sets are only the first layer of the structure carried by a homeomorphism. Because compactness is defined in terms of open covers, it is the next fundamental property to test for preservation. [quotetheorem:8839] A space may pass all compactness tests and still have different separation into pieces. The next invariant records whether a subset splits into two nonempty open parts. [quotetheorem:8840] A different kind of invariant concerns limits rather than subsets. The following statement records that every convergent net is carried to exactly the corresponding convergent net. [quotetheorem:8841] After seeing what homeomorphisms preserve, the relation among spaces itself needs structural control. A classification relation should permit identity maps, reversed identifications, and chains of identifications. [quotetheorem:8842] A common verification problem remains: continuous bijections are easy to find, while inverse continuity is harder. Compact domains and Hausdorff codomains solve this problem in many analytic examples. [quotetheorem:318] This theorem should not be mistaken for a general fact about all continuous bijections. The circle example above fails because the domain $[0,2\pi)$ is not compact. ## Relationship to Other Concepts Homeomorphism is the isomorphism notion in the category of topological spaces. The objects are topological spaces, the morphisms are continuous maps, and the isomorphisms are exactly the homeomorphisms. This categorical viewpoint explains why the definition requires the inverse map to be continuous: an isomorphism must be reversible inside the same structure-preserving class. In [Metric Space](/page/Metric%20Space) theory, homeomorphisms separate topological information from metric information. Two metrics on the same set may give the same topology, in which case the identity map is a homeomorphism between the resulting topological spaces. Isometries are stronger than homeomorphisms: they preserve distances, while homeomorphisms preserve only open sets and convergence. For Manifold theory, homeomorphisms describe when two spaces have the same underlying topological shape. Smooth manifolds require more structure: a diffeomorphism must be a homeomorphism whose coordinate expressions are smooth with smooth inverse. Thus homeomorphism is the topological layer beneath differentiable structure. In quotient constructions, homeomorphisms identify when the topology imposed by a quotient has produced the intended space. For example, identifying the endpoints of a closed interval produces a quotient space homeomorphic to the circle. This connects homeomorphism to [Quotient Topology](/page/Quotient%20Topology) and to many geometric constructions. Many arguments that spaces are not homeomorphic work by finding a feature that every homeomorphism must preserve and then checking that the two spaces disagree on that feature. To use this method cleanly, we isolate the preserved assignment itself. [definition: Topological Invariant] Let $\mathcal{C}$ be a class of topological spaces, and let $\mathcal{A}$ be a class whose elements can be compared by equality or isomorphism. A topological invariant on $\mathcal{C}$ with values in $\mathcal{A}$ is an assignment $I: \mathcal{C} \to \mathcal{A}$ such that, whenever $X, Y \in \mathcal{C}$ are homeomorphic, the assigned values $I(X)$ and $I(Y)$ are equal or isomorphic in $\mathcal{A}$. [/definition] The target class $\mathcal{A}$ may consist of truth values, numbers, groups, rings, vector spaces, or other structured objects, provided the relevant notion of agreement is specified. The essential requirement is that the invariant cannot distinguish between homeomorphic spaces. ## References [Topology](/page/Topology). [Continuity](/page/Continuity). [Compact Space](/page/Compact%20Space). [Connected Space](/page/Connected%20Space). James R. Munkres, *Topology* (2000). John L. Kelley, *General Topology* (1955). Stephen Willard, *General Topology* (1970).