Abstract
This chapter might seem odd in that it lists a huge number of topological properties and connections between them. What it shows is that the requirement that a manifold be metrisable is extremely versatile. We list over 100 conditions each of which is equivalent to metrisability of a manifold. At one extreme, metrisability of a manifold implies that it may be embedded as a closed subset of some Euclidean space while at the other extreme knowing that every open cover of the form \(\{U_{\alpha }\ /\ {\alpha }<{\omega }_1\}\) with \(U_{\alpha }\subset U_{\beta }\) whenever \({\alpha }<{\beta }\) has an open refinement which is point countable on a dense subset is sufficient to guarantee that a manifold is metrisable. Space precludes giving full details of the proofs. Instead we give brief ideas of the proofs and refer the interested reader to original sources for complete proofs. The content of this chapter is taken from [21].
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Addis, D.F., Gresham, J.H.: A class of infinite-dimensional spaces. Part I: Dimension theory and Alexandroff’s problem. Fund. Math. 101, 195–205 (1978)
Arhangel’ski\(\breve{\rm {i}}\), A.V., Buzyakova, R.Z.: On linearly Lindelöf and strongly discretely Lindelöf spaces, Top. Proc. 23, 1–11 (Summer 1998)
Arhangel’ski\(\breve{\rm {i}}\), A.V., Choban, M.M.: Compactly metrizable spaces and a theorem on generalized strong \(\varSigma \)-spaces. Top. Appl. 160, 1168–1172 (2013)
Babinkostova, L.: Selective screenability game and covering dimension. Top. Proc. 29(1), 13–17 (2005)
Cao, J., Gauld, D., Greenwood, S., Mohamad, A.: Games and metrisability of manifolds. N. Z. J. Math. 37, 1–8 (2008)
Cao, J., Junnila, H.: When is a Volterra space Baire? Top. Appl. 154, 527–532 (2007)
Cao, J., Mohamad, A.: Metrizability, manifolds and hyperspace topologies. JP J. Geom. Topol. 14, 1–12 (2013)
Caserta, A., Di Maio, G., Kočinac, L.D.R., Meccariello, E.: Applications of k-covers II. Top. Appl. 153, 3277–3293 (2006)
Cohen, M.M.: Local homeomorphisms of Euclidean space onto arbitrary manifolds. Mich. J. Math. 12, 493–498 (1965)
Deo, S., Gauld, D.: Boundedly metacompact or finitistic spaces (to appear)
Di Maio, G., Kočinac, L.D.R., Meccariello, E.: Selection principles and hyperspace topologies. Top. Appl. 153, 912–923 (2005)
Dow, A., Zhou, J.: On subspaces of pseudoradial spaces. Proc. Amer. Math. Soc. 127, 1221–1230 (1999)
Fearnley, D.L.: Metrisation of Moore spaces and abstract topological manifolds. Bull. Aust. Math. Soc. 56, 395–401 (1997)
Fell, J.: A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space. Proc. Amer. Math. Soc. 13, 472–476 (1962)
Feng, Z., Gartside, P.: Spaces with a finite family of basic functions. Bull. Lond. Math. Soc. 43, 26–32 (2011)
Forster, O.: Lectures on Riemann Surfaces. GTM 81, Springer, New York (1981)
Gartside, P.M., Mohamad, A.M.: Cleavability of manifolds. Top. Proc. 23, 155–166 (1998)
Gartside, P.M., Mohamad, A.M.: Metrizability of manifolds by diagonal properties. Top. Proc. 24, 621–640 (1999)
Gauld, D.: A strongly hereditarily separable, nonmetrisable manifold. Top. Appl. 51, 221–228 (1993)
Gauld, D.: Covering properties and metrisation of manifolds. Top. Proc. 23, 127–140 (1998)
Gauld, D.: Metrisability of manifolds, a developing survey found at http://arxiv.com/abs/0910.0885
Gauld, D.: Selections and metrisability of manifolds. Top. Appl. 160, 2473–2481 (2013)
Gauld, D.: Some properties close to Lindelöf (to appear)
Gauld, D., Greenwood, S.: Microbundles, manifolds and metrisability. Proc. Amer. Math. Soc. 128, 2801–2807 (2000)
Gauld, D., Greenwood, S., Piotrowski, Z.: On Volterra spaces III: topological operations. Top. Proc. 23, 167–182 (1998)
Gauld, D., Mynard, F.: Metrisability of manifolds in terms of function spaces. Houst. J. Math. 31, 199–214 (2005)
Gauld, D., Vamanamurthy, M.K.: Covering properties and metrisation of manifolds 2. Top. Proc. 24, 173–185 (Summer 1999)
Gauld, D., van Mill, J.: Homeomorphism groups and metrisation of manifolds. N. Z. J. Math. 42, 37–43 (2012)
Gruenhage, G.: Generalized metric spaces. In: Kunen, K., Vaughan, J. (eds.) Handbook of Set-Theoretic Topology, pp. 423–501. Elsevier, Amsterdam (1984)
Gruenhage, G.: The story of a topological game. Rocky Mountain J. Math. 36, 1885–1914 (2006)
Gruenhage, G., Ma, D.K.: Baireness of \(C_k(X)\) for locally compact X. Top. Appl. 80, 131–139 (1997)
Heath, R.W., Lutzer, D.J., Zenor, P.L.: Monotonically normal spaces. Trans. Amer. Math. Soc. 178, 481–493 (1973)
Kechris, A.S.: Classical Descriptive Set Theory. Springer, New York (1995)
Kister, J.M.: Microbundles are fibre bundles. Ann. Math. 2(80), 190–199 (1964)
Matveev, M.V.: Some questions on property (a). Q. A. Gen. Top. 15, 103–111 (1997)
Milnor, J.: Microbundles part I. Topology 3(Suppl. 1), 53–80 (1964)
Mohamad, A.M.: Metrization and semimetrization theorems with applications to manifolds. Acta Math. Hung. 83(4), 383–394 (1999)
Nyikos, P.: Various smoothings of the long line and their tangent bundles. Adv. Math. 93, 129–213 (1992)
Pears, A.R.: Dimension Theory of General Spaces. Cambridge University Press, Cambridge (1975)
Reed, G.M., Zenor, P.L.: A metrization theorem for normal Moore spaces. In: Stavrakas, N.M., Allen, K.R. (eds.) Studies in Topology, pp. 485–488. Academic Press, New York (1974)
Reed, G.M., Zenor, P.L.: Metrization of Moore spaces and generalized manifolds. Fund. Math. 91, 203–210 (1976)
Scheepers, M.: Combinatorics of open covers I: Ramsey theory. Top. Appl. 69, 31–62 (1996)
Tkačenko, M.G.: Ob odnom svoistve bicompactov (On a property of compact spaces). Seminar po obshchei topologii (A Seminar on General Topology), pp. 149–156. Moscow State University P. H., Moscow (1981) (Russian)
Williams, S.W., Zhou, H.: Strong versions of normality. General topology and its applications. In: Proceedings of the 5th NE Conference, New York 1989. Lecture Notes in Pure and Applied Mathematics, vol. 134, pp. 379–389. Marcel Dekker, New York (1991)
Worrell Jr, J.M., Wicke, H.H.: Characterizations of developable topological spaces. Can. J. Math. 17, 820–830 (1965)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media Singapore
About this chapter
Cite this chapter
Gauld, D. (2014). Edge of the World: When Are Manifolds Metrisable?. In: Non-metrisable Manifolds. Springer, Singapore. https://doi.org/10.1007/978-981-287-257-9_2
Download citation
DOI: https://doi.org/10.1007/978-981-287-257-9_2
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-287-256-2
Online ISBN: 978-981-287-257-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)