Connected (topology)/Equivalent conditions

This page is supposed to be transcluded, if you are arriving here from a search page see connected or disconnected

Equivalent conditions to being connected/disconnected

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

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TODO: Bottom of page 114 in Mendelson, something about closed sets and connectedness

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