Negative + でください

Question

Please note this post does not answer the question, it’s not a duplicate.

熱い時開けないでください

This is the warning messages on your car for engine coolant. But why is it not

熱い時開けなくてください,

Since it should be てform + ください？

で is the てform of だ, so could I see this sentence as 熱い時開けない事でください, since it’s always a noun that comes before だ？

Besides, I’ve also seen 熱いとき開けないこと being used as well.

Another supporting evidence is instead of そんな事はありません, some people say そんな事はないです, which, I think I can interpret as そんな事はない事です, right?

Is it valid, in these two cases, to put こと after the verb and before the で/です?

Answer 1

なくて and ないで are both valid て-forms but they have different functions. When used to connect clauses, なくて indicates cause and ないで attendant circumstances.

弟がドアを開けなくて困った。
I had a hard time because my brother didn’t open the door (for me).

ドアを開けないで返事をした。
I replied without opening the door.

The former is used when some action is actually not performed and that causes some consequences, whereas the latter describes a situation in which some action is kept unperformed while some other action is performed.

ないでください is used to ask someone to refrain from doing something. It’s like asking them to keep the action of the verb unperformed. (*) なくてください doesn’t make sense.

ないで is also used in ないでおく. Again (*) なくておく doesn’t make sense.

寒いので窓を開けないでおこう。
Since it is cold, let’s keep the window closed (lit. unopened).

cf. 暑いので窓を開けておこう。
Since it is hot, let’s keep the window open.

Answer 2

When you use 熱い時開けなくて, you mean: when it's hot, do not open it and

なくて = な（い） + く　（liaison) + て

while ないでください means: please do not