Difference between revisions of "Connectedness"

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Content obtained and/or adapted from:
 
Content obtained and/or adapted from:
 
* [http://mathonline.wikidot.com/connected-and-disconnected-metric-spaces Connected And Disconnected Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license
 
* [http://mathonline.wikidot.com/connected-and-disconnected-metric-spaces Connected And Disconnected Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license
 +
* [http://mathonline.wikidot.com/basic-theorems-regarding-connected-and-disconnected-metric-s Basic Theorems Regarding Connected and Disconnected Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license

Revision as of 11:14, 8 November 2021

Connected and Disconnected Metric Spaces

Definition: A metric space is said to be Disconnected if there exists nonempty open sets and such that and . If is not disconnected then we say that Connected. Furthermore, if then is said to be disconnected/connected if the metric subspace is disconnected/connected.

Intuitively, a set is disconnected if it can be separated into two pieces while a set is connected if it’s an entire piece.

For example, consider the metric space where is the Euclidean metric on . Let , i.e., is an open interval in . We claim that is connected.

Suppose not. Then there exists nonempty open subsets and such that and . Furthermore, and must be open intervals themselves, say and . We must have that . So or and furthermore, or .

If then this implies that (since if then which implies that ). So if . If then and so so . If then and so . If then . Either way we see that .

We can use the same logic for the other cases which will completely show that is connected.

Licensing

Content obtained and/or adapted from: