open the batting

keep a weather eye open

to maintain a background awareness of something; to remain alert to changes without it occupying one's full attention.

keep an eye open

To remain vigilant for someone or something.

keep one's ears open

to listen for important information.

keep one's options open

To not commit to a decision, to keep various options available.

kick at an open door

To carry out a redundant activity.

leave the door open

To maintain the possibility of something happening in the future.

open a can of whoop ass

A good-humored threat of physical harm.

open book

Something whose salient aspects are obvious or easily interpreted.