You might want to put your hat off inside. 「室内では帽子を脱いだほうがいいかもね。」
You might want to doは、簡単に言うとshould よりも柔らかい感じで 〜したら？と表現できます。
余談ですが、take my hat off to my teacherとすると「私の先生を尊敬する」という意味になります。
ご参考になれば幸いです。
The first sentence is quite casual. It is easy to forget that you are wearing a hat when you first step inside. If you would like to be a little sarcastic, you may prefer the second sentence. Of course it is not a serious question.
*You forgot to take off your hat.-A hat can be worn for fashion or to protect yourself from the sun. For women depening on the type of hat, some are not necessary to take off. But a gentleman always takes off his hat when indoors as a sign of respect. Indoors mean under a roof.
Example Sentences:
There's a hat rack by he door, you can hang your hat there.
Excuse me sir your hat please.
*You forgot to take off your hat.
（帽子を脱ぐの忘れてますよ）
-"hat"（帽子）はおしゃれで被ることも、日差しから守るためにかぶることも出来ます。
女性には帽子の種類によって脱ぐ必要のないものもあります。
でも、男性は敬意を示すために、いつも"indoors"（室内）では帽子は脱ぐものです。
"Indoors"とは屋根の下、室内と言う意味です。
【例文】
There's a hat rack by ｔhe door, you can hang your hat there.
（帽子掛けがドアの所にあります、あなたの帽子をかけるといいですよ）
Excuse me sir your hat please.
（恐れ入りますが、帽子をお願いします）
Now that we are indoors...Are you keeping your hat on?
Some folks just love to wear a hat..(.even indoors) That may come down to their personal choice:-D Otherwise they may simply have forgotten to take it off!
Try a gentle and kind reminder;-))
Now that we are indoors...Are you still keeping your hat on?
室内でも、ただ帽子をかぶっているのが好きな人がいます。個人の自由ですからね。そうでなければ、ただ帽子をとるのを忘れているだけでしょう。親切に聞くのを心がけましょう。
Now that we are indoors...Are you still keeping your hat on?
室内にいるけど、帽子をかぶったままじゃない？
Would you mind taking your hat off is a polite request,whereas are you keeping your hat on is a question that implies your dislike of wearing inside. Do you know you are still wearing your hat is a question that is not designed to be answered but more just to bring the fact to persons attention
Would you mind taking your hat off?
帽子をとってもらってもいいですか？
are you keeping your hat on（帽子をかぶっているの？）は、相手が帽子をかぶっていることを好ましく思っていない質問であるのに対し、これは、丁寧なお願いの表現です。 Do you know you are still wearing your hat （まだ帽子をかぶっているの知ってる？）は、答えを求めるだけでなく、相手の注意を引く質問です。
You've forgotten to remove your hat, mr/ms./miss/mrs.
"Please remove your hat, sir/madame." This is a polite way of asking the person to remove (take off) their hat.
"You've forgotten to remove your hat, mr/ms./miss/mrs." Forgotten means to have forgot something. And this is just another way of asking an individual to please remove their hat.
"Please remove your hat, sir/madame."（帽子をお取りください）
- 帽子を取るようお願いする丁寧な言い方です。
"You've forgotten to remove your hat, mr/ms./miss/mrs."（帽子を取り忘れていますよ）
- "forgotten" は「忘れている」という意味です。これも同様に、帽子を取ってくださいとお願いしています。
It may be politer to ask why they are still wearing the hat, as they may not want to take it off. They may really like the hat, or they may still be feeling cold, so by asking the question you are able to see the reason behind why the hat is still on before insisting it is removed.
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