1. after time 是个成语,意思是:“迟到的 ”“超过规定时间的”，“过期的”

We cannot wait for anyone who arrives after time.

我们绝不等待任何迟到的人。

The publican was fined for selling beer after time.

这位酒店老板因出售过期啤酒而被罚了款。

2. after a time的意思则是“过了一段时间，过了很久”

After a time they came to a pond.

过了好久，他们来到一个池塘。

I didn't care for the place at first, but after a time I got to like it.

起初我不喜欢这个地方,但过了一段时间我便喜欢上它了。

He grew to like her after a time.

一段时期后，他渐渐喜欢起她来了。

After a time, you get to realize that these things don't matter.

过段时间你会觉得这些事情并不要紧。

After a time, the dog stopped following me.

过了一会儿，那条狗便不再跟着我了。