Closure and Interior of a set in a Topological space

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Open set(so closed set) are building blocks for a Topological space.  A random subset of a topological space (X, τ) need not be either open or closed but there is always an open and closed set associated with it.

Let A  X where (X, τ) is a topological space. We know that Φ is an open set i.e. contained in every subset of X. Can we go for an open set larger than Φ, contained in A? Similiarly, the smallest open set in which contain A. Same thing we can do for the closed set, largest and smallest set respectively contained and containing A. These questions give rise to the concept of Closure and Interior.

Closure of A  X  -: The closure of A in (X, τ), is defined as the smallest closed set, which contains A. For example, closure of [0,2) is [0,2] and closure of set S = {1/n : n  } is S  {0} with usual topology on . 
Some of the text use the following definition of closure as A  A', which is not an intuitive way of defining a new concept. Of course, it is equivalent to our original definition.

Interior of A  X  -: The interior of A in (X, τ), is the largest open set contained in A. Equivalently, it equals the union of all open sets contained in A. For example, interior of [0,2) is (0,2) and interior of  is Φ w.r.t. usual topology on .

Now take a look on following questions. 

What about the largest closed set contained in A ? What about the smallest open set containing A ?
Are these two questions are answerable from closure and interior ?

To be continued soon...

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