By I Gerald Bowen; U.S. Atomic Energy Commission. Civil Effects Test Operations.; et al
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Extra info for A model designed to predict the motion of objects translated by classical blast waves
Each set has a complement with two 3. 11 components, but one component of the complement of X is a disc, whereas both components of the complement of Yare cylinders, which we later prove cannot be homeomorphic to a disc. 3, the circles X and Yare non-equivalent in the cylinder. 20, again excluding the edge points. 15, a Mobius band is obtained by gluing together the ends of the rectangle, but first giving a half twist to one end. 21. 22: the circle goes round no times, once or twice, although we are not relying on the idea of "going round" for proof of non-equivalence.
We first show that f(8) belongs to Y, so we consider any neighbourhood N of f(~). Because f is continuous the pre-image M of N is a neighbourhood of 8. But 8 is in X, so there is some point x common to M and X. It follows that f(x) belongs to Nand Y. Hence f(8) is in Y, so that f(X) ~ Y. Similarly, f-l(y) ~ X, and we deduce that f(X) = Y. Thus X and Yare equivalent in S. This completes the proof. 7, and let W be ]0,00[, whose non-equivalence to X, Y, Z can now be shown. The closures of X, Y, Z and W have respectively 2,00,0,1 not-cut-points, and so are non-homeomorphic.
Thus D has a circuit, but A, B, C do not. A tree is a connected graph with no circuits. So A and B are trees, whereas C and D are not. Analogously to the way that homeomorphic sets are regarded as the same, graphs with identical structure-isomorphic graphs-are regarded as the same. By an isomorphism between graphs we mean bijections from the vertices and edges of one graph to those of the other such that a pair and its image pair are either both related or both unrelated. Graphs are isomorphic if there is an isomorphism between them.
A model designed to predict the motion of objects translated by classical blast waves by I Gerald Bowen; U.S. Atomic Energy Commission. Civil Effects Test Operations.; et al