Vipul: Created page with '{{topospace metaproperty satisfaction| property = connected space| metaproperty = continuous image-closed property of topological spaces}} ==Statement== Suppose $X$ …'

2009-12-23T06:32:58Z

Created page with '{{topospace metaproperty satisfaction| property = connected space| metaproperty = continuous image-closed property of topological spaces}} ==Statement== Suppose <math>X</math> …'

New page

{{topospace metaproperty satisfaction|
property = connected space|
metaproperty = continuous image-closed property of topological spaces}}

==Statement==

Suppose <math>X</math> is a [[connected space]] and <math>f:X \to Y</math> is a [[continuous map]]. Then, <math>f(X)</math>, endowed with the [[subspace topology]] from <math>Y</math>, is a [[connected space]].

==Related facts==

===Weaker facts===

* [[Stronger than::Connectedness is coarsening-preserved]]

===Other related facts===

* [[Path-connectedness is continuous image-closed]]

==Applications==

* [[Intermediate-value theorem]]: This states that any continuous real-valued function on a connected space that takes the real values <math>a</math> and <math>b</math> with <math>a < b</math>, takes all real values in the interval <math>[a,b]</math>.

==Proof==

{{fillin}}

Connectedness is continuous image-closed - Revision history

Vipul: Created page with '{{topospace metaproperty satisfaction| property = connected space| metaproperty = continuous image-closed property of topological spaces}} ==Statement== Suppose X …'

Vipul: Created page with '{{topospace metaproperty satisfaction| property = connected space| metaproperty = continuous image-closed property of topological spaces}} ==Statement== Suppose $X$ …'