How can I support a boyfriend who asked for some space when he has troubles at work?

Well, respect his wishes and give him space. This doesn't necessarily mean not talking to each other at all but in this scenario I would advise you to let him take this time for himself. I don't really know what's going on with him but it seems that he has a lot on his mind right now and might be affecting his mental health. Let him be the one approaching you do not force anything. Be supportive. You can text him something sweet and simple once a day like "I hope you're okay. I'm always here if you need me" or just a simple "I miss you" text message. Good luck.
Give him the space without invading it. Look out for the other girl without challenging him.
I don't think looking for someone else is a good idea
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You can support him by just respecting his wishes and giving him space like he asked for
For how long
Until he’s ready to talk. Don’t pressure him.
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