testing for continuity via closure operation
. Use the identity for any function . Then . So if is closed (or open), is open (or closed), whence is closed (or open).
. Suppose first that is continuous. Since
, which is closed in . So , and therefore . As a result,
Conversely, let be closed in . Then . Let . So . Let . Then . So . As a result, is closed.
. First, assume . Let and . So . Then . As a result, .
Conversely, assume . Let and . So . Then
. First, assume . We use the identity: . Then
Conversely, assume . We use the identity . Then
|Title||testing for continuity via closure operation|
|Date of creation||2013-03-22 19:09:11|
|Last modified on||2013-03-22 19:09:11|
|Last modified by||CWoo (3771)|