We know space-time in General Relativity locally looks like (Topologically is homeomorphic to) Minkowski space-time which its topology may be Zeeman topology, not E^4 (the space R^4 with open balls topology defined by Euclidean metric). On the other hand, most of mathematics and physics text books start Manifold chapters by introducing Topological Manifolds which locally look like E^4 then after, they define extra structures on them and introduce Differentiable Manifolds, Metric, Riemannian and Pseudo Riemannian Manifold.... Hence, it seems by definition, our space-time is not a Topological Manifold! Then on what kind of things we are working?!”
R^4 is not a topology, but a space. It can be endowed with various topologies, including the Zeeman one, and thus becomes a topological space. The Zeeman topology is finest than the ball one, that is, every ball is an open set of the Zeeman topology. As a manifold is locally homeomorphic to R^n, its induced topology is whatever topology is defined on R^n.
Dear Amir,Topology definition, basically originated from set theory, briefly a collection of subsets of a set A, call it σ , with some trivial properties usually is defined as topology. The set A and its topology σ constitute a topological space when the members of A are point. Thus the topology of sets of points studies the specific relations of considered points with each other, which remains unchanged when we impose a continuous deformation on this set (space). for instance the topology of sphere and torus is not the same but topologies of a "cup" and " doughnut" are the same. Moreover when we assume the space-time in terms of General relativity we impose a metric and define a metric space with some properties, this space is also a topological space with its features of neighborhoods, tangent spaces and coordinate charts, and therefor is a manifold ( a topological space with Euclidean space property at infinitesimal neighborhood of every points). In this accepted view your conclusion: "our space-time is not a Topological Manifold! " does not seem meaningful.
The statement about general relativity and Minkowski spacetime is incorrect: Minkowski spacetime is by no means the only solution to Einstein's equations. By construction the spacetimes defined by the solutions to Einstein's equations are metric spaces-since the metric is what's being solved for. There are then theorems that imply certain results about the topological properties of the spacetimes. But the statement about ``our spacetime'' (which, in fact, isn't Minkowski-the Universe is expanding, as we now know) being, or not, a ``topological manifold'' is free of content, absent further details. Our spacetime does seem to be described very well as a manifold and its topological aspects, that are compatible with the metric being a solution of Einstein's equations, seem to be understood. All this is described in detail in textbooks on general relativity.
We need a topology in order to define continuous functions, such as transition functions and fields. For this purpose, many different topologies work as well, since they have as good as no physical significance. The metric gives a much more precise notion of proximity.
Spacetime topology is orientable, homeomorphic, paracompact, Housdorff and continuous and is defined in a differentiable manifold.
The answers by @RobertLaw and @ClaudePierreMassé are very much to the point. There is, however, the crucial question how an open set can be operationally defined in terms of measurable quantities.
To do this, it is enough to define a basis of open sets. of an open set. Define a basis open set as the intersection of a forward timeline cone of a point (i.e., the part of space time that lies in the future that can be causally influenced by that point) with a backward timeline cone in an other point (i.e. the part of space time in the past that can influence the other point). Thus the topology is defined in terms of the causal structure.
Please see this useful article:
"On the Topology of Lorentzian manifolds"
Renee Hoekzema
March 13, 2011
Thinking topologically gets you tied up in knots. The correct representation of Minkowski space uses hyperbolic (Lobachevsky) geometry as Minkowski himself observed (1907). Hyperbolic geometry is not talked about by topologists. More precisely 4 dimensional space-tme is the Cartesian product of 3 dimensional hyperbolic geometry with a 1 dimension real line (proper time).
Topological properties are independent of, and more primitive than, metrical properties. R4 is a continuous topological space, not a metric space.
Considerable confusion is created in many textbooks by introducing the topology of Rn by inappropriately imposing a Euclidean metric on it and then talking about the topology in terms of “open balls”. That is not required, and is very misleading. R4 is simply the set of all quadruples of real numbers (x1, x2, x3, x4). It already has a natural topology (ie., continuity properties). It doesn’t have, or need, a “metric”. R4 and E4 are not the same thing!
In differential geometry an n-dimensional topological manifold is defined as a Hausdorff topological space every point of which is contained in an open set that is homeomorphic to an open set of Rn. “Spacetime” is a 4-dimensional manifold in that sense. It looks locally like R4 (not E4!).
Spacetime is a 4-dimensional topological manifold with additional structure – a metric field that is locally Minkowskian.
Charles ~
The confusion between Rn and En is very common in topology textbooks - and on web sites. That's why Amir felt the need to post the question. You will also recall my recent quibble with Luiz on this same point. To back up what I just said I did a casual bit of Googling and came up rapidly with the following examples:
https://books.google.co.in/books?id=ZQVGAAAAQBAJ&pg=PA38&lpg=PA38&dq=manifold+open+ball
http://math.stackexchange.com/questions/426658/what-is-the-difference-between-a-ball-and-a-neighbourhood
http://www.math.harvard.edu/~elkies/M55a.02/pdflatex/top2.pdf
http://en.wikipedia.org/wiki/Ball_%28mathematics%29
http://www.maths.bris.ac.uk/~maxmr/opt/closedandconvex.pdf
http://www.personal.psu.edu/axk29/TOPOLOGY/Chapter1.pdf [top of page 6: "Euclidean space Rn"(!).]
http://www.math.uiuc.edu/~lerman/518/f11/top.pdf
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