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Topological Space : Subspace

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Let X be a topological space and let Y be a subset of X. Check that the so-called subspace topology is indeed a topology on Y.

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Definition of Topological Space: Suppose X is any non-empty set. Let T be a set of subsets of X such that the following are satisfied.
i) The empty set is in T i.e. and the set X is in T i.e.
ii) Arbitrary union of elements of T is an element of T i.e. where I is an index set and
iii) Finite intersection of elements of T is an element of T i.e. where , ( - the set of natural numbers) and

Then we say that ...

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