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  "body": "\n## Summary\n\nAn isomorphism of topological groups is a group isomorphism that is also a homeomorphism.\nTwo isomorphic topological groups are the same in every respect that involves both the group law and the topology.\nA continuous bijective homomorphism need not be an isomorphism, because its inverse may fail to be continuous.\nThe open mapping theorem for locally compact groups gives conditions under which this cannot happen.\n\n## Explanations\n\n### L1 human\n\nTwo topological groups are isomorphic when one can rename the elements of one to get the other, keeping both the multiplication and the notion of nearness.\nFor example, the positive real numbers under multiplication and the real numbers under addition are isomorphic.\nThe logarithm turns products into sums, and both the logarithm and the exponential function are continuous.\nSo these two groups are copies of each other, even though they look different at first.\n",
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