A String theory and locality of small dimensions

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- I have in essence never understood the concept of curled up dimensions in string theory ; in the sense that if it occurs in this universe isn't that included in our 3 dimensional space ? My question is more fundamental in the sense from a topology viewpoint : was it correct to involve curled up dimensions , because there remains the 'surrounding environment'. And or did string theorists in the end abuse geometry from an 18th century concept of dimensions. Tx for your insights,

M.
 
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For a visualization, consider the surface of a cylinder. It is curled in one direction, but flat in another.
 
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