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Connectedness tells us when a space is made of one piece, but many spaces encountered in [Topology](/page/Topology), [Metric Space](/page/Metric%20Space), and [Continuity](/page/Continuity) are not globally connected. The next natural question is local-to-global in spirit: if the whole space splits apart, what are its largest indivisible pieces? Connected components answer that question. They decompose a [topological space](/page/Topological%20Space) into maximal connected regions, giving a canonical partition that is independent of coordinates, metrics, or chosen paths.

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The connected component of a point is especially useful because it turns a global property into a pointwise invariant. In analysis, components describe domains of definition for differential equations, isolate pieces of open subsets of Euclidean space, and control where continuous images can move. In algebraic topology, they are the first layer of homotopical information, recorded by the set of path components and compared with connected components through local path connectedness.

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## Definition

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The pointwise problem is to find the whole connected region belonging to a chosen point $x$. Small connected subsets such as $\{x\}$ give too little information, while arbitrary connected subsets through $x$ are not canonical. Taking all connected subsets through $x$ at once produces the largest connected piece forced by the topology.

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[definition: Connected Component] Let $(X, \tau)$ be a topological space and let $x \in X$. The connected component of $x$ in $X$, denoted $C_x$, is \begin{align*} C_x := \bigcup \{A \subset X : A \text{ is connected and } x \in A\}. \end{align*} [/definition]

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This definition is primary, but it depends on the ordinary notion of a connected subset. That supporting notion is stated next so the component formula can be read without ambiguity.

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[definition: Connected Subset] Let $(X, \tau)$ be a topological space. A subset $A \subset X$ is connected if there do not exist nonempty subsets $U,V \subset A$ such that $A = U \cup V$, $U \cap V = \varnothing$, and $U,V$ are open in the [subspace topology](/page/Subspace%20Topology) on $A$. [/definition]

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The definition uses the subspace topology because a subset should be judged using the open sets it inherits from the ambient space. This makes connectedness intrinsic to $A$ once $A \subset X$ is fixed, and it explains why the union defining $C_x$ searches among connected subsets of $X$.

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[remark: Empty and Singleton Subsets] The empty subset and every singleton subset of a topological space are connected. For a singleton, there is no way to split its only point between two nonempty disjoint pieces. [/remark]

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Since $\{x\}$ is connected, the indexing family in the definition of $C_x$ is nonempty and $x \in C_x$.

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A decomposition is useful only if its pieces can be named independently of the particular point used to find them. Since many different points may produce the same set $C_x$, we also need language for the resulting subsets themselves.

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[definition: Connected Component of a Space] Let $(X, \tau)$ be a topological space. A subset $C \subset X$ is a connected component of $X$ if there exists $x \in X$ such that $C = C_x$. [/definition]

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The two definitions are tied together by the pointwise notation: the connected components of $X$ are exactly the distinct sets among the family $(C_x)_{x \in X}$.

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## Equivalent Characterisations

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The union formula is constructive, but in practice one often recognises a component by a maximality property. This shifts attention from the point $x$ to the connected subset itself.

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[quotetheorem:302]

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The theorem justifies the common phrase "maximal connected subset." The subtle point is that a union of connected sets need not be connected in general, but it is connected when all the sets in the union share the point $x$. This motivates isolating maximal connectedness as its own recognition criterion, since many arguments identify components by proving that a connected set cannot be enlarged.

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[definition: Maximal Connected Subset] Let $(X, \tau)$ be a topological space. A subset $C \subset X$ is a maximal connected subset of $X$ if $C$ is connected and whenever $D \subset X$ is connected with $C \subset D$, one has $D = C$. [/definition]

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The preceding theorem says that connected components and maximal connected subsets are the same objects. The definition is often better for recognition, while the union definition is better for construction. Its limitation is that maximality names the pieces one at a time; it does not yet explain how all points of $X$ are sorted into those pieces.

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The partition problem asks when two points should belong to the same connected piece. A relation is the right language for this question: it can group points by the existence of a connected subset containing them both, and then its equivalence classes can be compared with the sets $C_x$. This relation is introduced so that the component decomposition can be treated as a genuine partition, not merely as a collection of large connected subsets.

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