# Lower semicontinuous functions

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Let be a metric space (or more generally a topological space). A function is **lower semicontinuous** if

is open in for all .^{[1]}

## Related Properties

- If is lower semicontinuous and the is lower semicontinuous.
^{[2]}

- If is a topological space and is any open set, then is lower semicontinuous.
^{[2]}

- If are lower semicontinuous, then is lower semicontinuous.
^{[2]}

- If is a locally compact Hausdorff space, and is lower semicontinuous, then where denotes the space of all continuous functions on with compact support.
^{[2]}

- If is an convergent sequence in converging to some , then .
^{[1]}

- If is continuous, then it is lower semicontinuous.
^{[1]}

- In the case that , is Borel-measurable.
^{[3]}

- If is a collection of lower semicontinuous functions from to , then the function is lower semicontinuous.
^{[4]}

## Lower Semicontinuous Envelope

Given any bounded function , the lower semicontinuous envelope of , denoted is the lower semicontinuous function defined as

## References

- ↑
^{1.0}^{1.1}^{1.2}Craig, Katy.*MATH 201A HW 1*. UC Santa Barbara, Fall 2020. - ↑
^{2.0}^{2.1}^{2.2}^{2.3}Gerald B. Folland,*Real Analysis: Modern Techniques and Their Applications, second edition*, §7.2 - ↑ Craig, Katy.
*MATH 201A HW 4*. UC Santa Barbara, Fall 2020. - ↑ Craig, Katy.
*MATH 201A HW 5*. UC Santa Barbara, Fall 2020.