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Let us now set aside the question of the submicroscopic structure of space-time and concentrate, instead, on its large-scale properties. In this case, we may imagine that the smooth manifold picture will be adequate and that its structure in the large can be obtained by piecing together smaller "locally Euclidean" patches, in the manner of overlapping coordinate neighborhoods of differential Geometry. Thus we might arrive at a topology for space-time, in the large, different from a Euclidean topology. Unfortunately too little is known about the large-scale structure of the universe to enable us to make any statement with confidence concerning its global topology (apart, perhaps, from certain statements about its orientability). Thus, it might be that the topology of space-time on a large scale is not at all interesting.

Battelle Recontres, Sir Roger Penrose, page-123.

Could it be explained what exactly Penrose means when he says the topology of space-time? How can one picture it? I am assuming when he says Euclidean topology, he is talking about the metric ball topology in $\mathbb{R^4}$.

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This is a very common abuse of language among physicists, topologists and geometers: When they talk about "the (global) topology" of a space, they don't really mean the technical meaning of a topology as a collection of open sets or whatever.

Instead, what people mean are answers to questions like "Does this space have holes?" (homology), "Does this space have non-trivial loops?" (fundamental group), "Can you embed this space faithfully in $n$ dimensions?", i.e. properties of the space that are not "local", that you cannot see if you only look at infinitesimal neighbourhoods of points or a single coordinate chart, but that concern the full space.

In similarly vague words, the "topology" of a space in this sense is all the information that makes up its shape, that tells you how, qualitatively, a sphere is different from a donut or the figure 8. However - and that's where this abuse of language comes from - all this information is already included in the technical information of the topological space: It can be computed directly (if laboriously) from the technical meaning of the topology as a collection of open sets.

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  • $\begingroup$ So, what does euclideanness mean in the sense of the quantities you mentioned? $\endgroup$ May 22 at 21:02
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    $\begingroup$ @Aplateofmomos Just that every coordinate patch has the topology of $\mathbb{R}^n$. $\endgroup$
    – ACuriousMind
    May 22 at 21:26

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