Disconnected (topology)

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Note: much more information may be found on the connected page, this page exists just to document disconnectedness.

Definition

A topological space, $(X,\mathcal{ J })$, is said to be disconnected if:

• $\exists U,V\in\mathcal{J}[U\ne\emptyset\wedge V\neq\emptyset\wedge V\cap U=\emptyset\wedge U\cup V=X]$, in words "if there exists a pair of disjoint and non-empty open sets, $U$ and $V$, such that their union is $X$"

In this case, $U$ and $V$ are said to disconnect $X$ and are sometimes called a separation of $X$.

Disconnected subset

Let $A\in\mathcal{P}(X)$ be an arbitrary subset of $X$ (for a topological space $(X,\mathcal{ J })$ as given above), then we say $A$ is disconnected in $(X,\mathcal{ J })$ if:

• $A$ is a disconnected topological space when considered with the subspace topology (from $(X,\mathcal{ J })$)

Equivalent conditions

To a topological space $(X,\mathcal{ J })$ being connected:

• A topological space is connected if and only if the only sets that are both open and closed in the space are the entire space itself and the emptyset
  • Some authors give this as the definition of a connected space, eg
• A topological space is disconnected if and only if there exists a non-constant continuous function from the space to the discrete space on two elements
• A topological space is disconnected if and only if it is homeomorphic to a disjoint union of two or more non-empty topological spaces

To an arbitrary subset, $A\in\mathcal{P}(X)$, being connected:

• Obviously, the only sets being both relatively open and relatively closed in $A$ are $\emptyset$ and $A$ itself. (This comes directly from the subspace definition above)
• A subset of a topological space is disconnected if and only if it can be covered by two non-empty-in-the-subset and disjoint-in-the-subset sets that are open in the space itself
  • Then apply the definition above, a subset is considered connected if it is not disconnected
• A subset of a topological space is connected if and only if and only if the only two subsets that are both relatively open and relatively closed with respect to the subset are the empty-set and the subset itself

See also

• Connected - a space is connected if it is not disconnected
  • Much more information is available on that page, this is simply a supporting page