By Chern S.S., Li P., Cheng S.Y., Tian G. (eds.)

ISBN-10: 9810223854

ISBN-13: 9789810223854

Those chosen papers of S.S. Chern speak about subject matters similar to imperative geometry in Klein areas, a theorem on orientable surfaces in 4-dimensional house, and transgression in linked bundles

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**Additional resources for A Mathematician and His Mathematical Work: Selected Papers of S S Chern**

**Example text**

Where Ä D maxfÄ1 ; : :L : ; Än g. 265. Xt /. Deduce from this fact that L l l if Xt Yt for any t 2 T then X Y D t2T Yt . 266. Suppose L that a space Ji is homeomorphic to I for any i D 1; : : : ; n and let J D fJi W 1 Ä i Ä ng. Prove that the space J ˚ D is l-equivalent to I for any finite space D. Deduce from this fact that (i) connectedness is not preserved by l-equivalence; (ii) for any cardinal Ä there exist l-equivalent spaces X and Y such that X has no isolated points and Y has Ä-many isolated points.

3 Linear Topological Spaces and l-Equivalence 27 201. Prove that the topology of any linear topological T0 -space is Tychonoff. 202. Let L be a linear topological Tychonoff space. Prove that, for any local base B of the space L at 0, the following properties hold: (1) (2) (3) (4) (5) for any U; V 2 B, there is W 2 B such U \V; T that W every B 2 B is an absorbing set and B D f0g; for any U 2 B, there exists V 2 B such that V C V U; for any U 2 B and x 2 U , there exists V 2 B such that x C V U; for any U 2 B and " > 0 there is V 2 B such that V U for any 2 .

Z/. 185. Suppose that n 2 N and a space Xi is metrizable for every i Ä n. Prove that, for any countable ordinal 2, (i) if Xi 2 A for all i Ä n then X1 : : : Xn 2 A ; (ii) if Xi 2 M for all i Ä n then X1 : : : Xn 2 M . 186. X / \ M˛ for every n 2 !. Prove that X 2 M˛ . 187. g and Xn 2 M n for every n 2 !. 188. Given a countable ordinal 2, let M be the class of absolute Borel sets of multiplicative class . X / W ˛ < n g. 189. X / W ˛ < n g. 190. Prove that any analytic space has a complete sequence of countable covers.

### A Mathematician and His Mathematical Work: Selected Papers of S S Chern by Chern S.S., Li P., Cheng S.Y., Tian G. (eds.)

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