Essential infimum and essential supremum

Okay, so imagine you have a basket of apples. Some of the apples may be really big, some may be really small, and some may be just right. Now, we want to find the smallest and largest sizes of apples in the basket, but there's a catch - some of the apples may be rotten or have worms in them, so we don't want to include those when we are finding the smallest and largest sizes.

The smallest size of non-rotten apple is called the essential infimum, and the largest size of non-rotten apple is called the essential supremum.

To find the essential infimum, you have to look at all the non-rotten apples and find the smallest one. Even if there are smaller rotten apples, we don't count those. We only look at the non-rotten ones. So if the smallest non-rotten apple is still pretty big, then that would be the essential infimum.

To find the essential supremum, you have to look at all the non-rotten apples and find the largest one. Even if there are bigger rotten apples, we don't count those. We only look at the non-rotten ones. So if the largest non-rotten apple is still pretty small, then that would be the essential supremum.

So, essentially, the essential infimum and essential supremum are like finding the smallest and largest things in a group, but only considering the non-rotten or non-bad options.