Define dense set and an example of dense subset in usual topology. – Essaylink

Question:1. Define dense set and an example of dense subset in usual topology. Show that – = – ° (1+1+4)
Question:2.Consider the topology on = { , , , , } defined by
= {Ø, , { }, {,}, {, ,}, { , , , }, { , , }}
Determine the derived sets of = { , , } and = { }. (5)
Question:3. What do you mean by neighborhood system? Show that a set is open if and only if it is neighborhood of each of its elements. (2+5)
Question:4. In a topological space ,, show that if ? then show that ° ? ° and ¯ ? . (6)
Question:5. What is the difference between a bases and a subbases? Let be a base for the topological space , and * be a class of open sets containing . Show that *is also a base for ,. (2+4)
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