# MTH402 Solutions

MTH402 Tma Solutions

1. Let X be a topological space. Then one of the following conditions does not hold

\(\phi\) and X are closed

Arbitrary intersection of closed sets is closed

—>> Infinite unions of closed sets are closed

Finite unions of closed sets are closed

2. Let \(\mathbb R\) be with the usual standard topology and let A \subsets \mathbb R\).Then A is open in \(\mathbb R\) if there exists an interval I such that I\subset A. For a,b\(\epsilon\mathbb R, I =

—>> I = ( a, b)

I = ( a, b]

I = [ a,b]

I = [a,b)

\(\mathbb B’ = {[a,b] : a,b\epsilon\mathbb R; a<b}\)

\(\mathbb B’ = (a,b) : a,b\epsilon\mathbb R; a<b\)

10. A metric on a set X with a function d : X x X \(\rightarrow\mathbb R\) holds for all but one property in the following:

d(x,y)\(\geq 0\forall x,y\epsilon X\)

\(d(x,y) = d(y,m)\forall x,y\epsilon X\)

\(d(x,y)\leq d(x,y) + d(y,z)\forall x,y,z\epsilon X\)

—>> \(d(x,y) = 0 \)whenever \(\neq\) and \(x,y\epsilon X\)

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