The universe is expanding. Into what ?
The math model of expansion does not need any surround to expand "into".
BTW the math model of the cosmos can be curved without having to sit in some surrounding space to be curved "in".
In 1850 a young guy in Göttingen named Bernie discovered the math tools needed to define and measure both spatial curvature and expansion *internally* without the need for surroundings. You just work within the space itself. You do geometry intrinsically.
So the upshot is that cosmology is a mathematical science---you fit data to math models. It is not a verbal science. So when they say "expands", the word is only the best available English language approximation for the idea. You can't automatically go by words, you have to ask what is really meant. Dig down to the underlying mathematical meaning.
There could be a surround, but since there is no evidence of one, it is more a speculative fairytale fantasy. Cosmologists normally do not assume any surrounding space in which our universe is embedded because there is no indication that one exists, nor is there any conceptual need to assume one.
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