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Topology2010aug - Topology Qualifying Exam Do all of the...

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Topology Qualifying Exam August 23, 2010 Do all of the following problems. 1. Let X be a topological space. (a) (5 points) Define what it means to say that X is regular . (b) (10 points) Give a complete proof that a product of two regular spaces is regular. 2. Let X be a space. (a) (5 points) Define what it means to say that X is normal . It is easy to show that a normal space is regular. On the other hand the space R 2 l , the so-called Sorgenfrey plane , is a space that is regular, but not normal. Here R l is the real line with the lower limit topology. (b) (5 points) Prove that R 2 l is regular. (c) (10 points) Prove that R 2 l is not normal. 3. Let X be a topological space. (a) (5 points) Define what it means to say X is metrizable . (b) (10 points) Let R be the real line with the usual topology. Let R ω be the product of countably many copies R and give R ω the product topology. Show that R ω is metrizable. 4. Let X be a topological space. (a) (10 points) Give two necessary conditions, other than separation properties, for X to be metrizable.
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