Editing Connected space (section)

== Theorems <!--'Main theorem of connectedness' redirects here-->==
*'''Main theorem of connectedness'''<!--boldface per WP:R#PLA-->: Let <math>X</math> and <math>Y</math> be topological spaces and let <math>f:X\rightarrow Y</math> be a continuous function. If <math>X</math> is (path-)connected then the image <math>f(X)</math> is (path-)connected. This result can be considered a generalization of the [[intermediate value theorem]].
*Every path-connected space is connected.
*In a locally path-connected space, every open connected set is path-connected.
*Every locally path-connected space is locally connected.
*A locally path-connected space is path-connected if and only if it is connected.
*The [[closure (topology)|closure]] of a connected subset is connected.  Furthermore, any subset between a connected subset and its closure is connected.
*The connected components are always [[closed set|closed]] (but in general not open)
*The connected components of a locally connected space are also open.
*The connected components of a space are disjoint unions of the path-connected components (which in general are neither open nor closed).
*Every [[Quotient space (topology)|quotient]] of a connected (resp. locally connected, path-connected, locally path-connected) space is connected (resp. locally connected, path-connected, locally path-connected).
*Every  [[product topology|product]] of a family of connected (resp. path-connected) spaces is connected (resp. path-connected).
*Every open subset of a locally connected (resp. locally path-connected) space is locally connected (resp. locally path-connected).
*Every [[manifold]] is locally path-connected.
*Arc-wise connected space is path connected, but path-wise connected space may not be arc-wise connected
*Continuous image of arc-wise connected set is arc-wise connected.