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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.

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It does not remember lengths, angles, differentiability, or coordinates.

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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).

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In analysis this distinction matters constantly.

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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.

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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.

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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.

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## Definition

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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.

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The inverse must also respect the topology.

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This two-sided condition is needed because topological sameness should allow arguments to move from $X$ to $Y$ and then return without losing information.

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[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]

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The definition treats $f$ and $f^{-1}$ symmetrically.

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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 $Y$.

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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.

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[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]

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When $X$ and $Y$ are homeomorphic, we often write that $X \cong Y$ in a topological context.

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The symbol must be read from context, since it can also denote other kinds of isomorphism in algebraic settings.

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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.

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[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]

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