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)
3. A set is nowhere dense if the set \(bar {A}\) has empty ____________
Inferior
Accumulation point
—>> Interior
Exterior
4. When is \(B\) aneuclidean topology \(\mathbb R?\) When
\(\mathbb B = (a,b): a,b\epsilon\mathbb R, a=b\)
\(\mathbb B = (a,b): a,b\epsilon\mathbb R, a>b\)
—>> \(\mathbb B = (a,b): a,b\epsilon\mathbb R, a<b\)
\(\mathbb B = (a,b): a,b\epsilon\mathbb R, a/b\)
5. Let X be a set. A topology on X is acollection \(\tau \)of subsets of X, for which one of these does not hold:
The set X itself and the empty set \(\phi\) are in \(\tau\)
Arbitrary unions\( \bigcup_{}^{}\cup\) of elememnts of \(\tau\) are in \(\tau\)
Finite intersection \(\bigcap\cup_k\) of elements of \(\tau\) are in \(\tau\)
—>> The set X x X is also a member of \(\chi\)
6. A set \(\bigcup\) is open in the meric topology induced by d if and only for each x\(\epsilon\bigcup\), there exist \(\epsilon> 0\) such that
—>> B_d( x,\(\epsilon)\subset\bigcup\)
\(B_d( x,\epsilon)\supset\bigcup\)
\(B_d( x,\epsilon) = \bigcup\)
\(B_d( x,\epsilon)\) > \bigcup\)
7. The countable collection B = { ( a, b ) : a<b, a,b\(\epsilon\mathbb Q\)} is a ___________________________ for a topology on \(\mathbb R\)
Platform
Nucleus
—>> Basis
Reason
8. Let \(\pi_{1}( x, y) =x\) and \(\pi_{2}( x,y) =y \)then \(\pi_{1} : X x Y\rightarrow X \)and \(\pi_{2} : X x X\rightarrow\) Y. The maps \(\pi_{1}\) and \(\pi_{2}\) are called ____________________________
—>> Projections of X x X
Projections of X x X
Projections of Y x Y
projections of X^2 x Y^2
9. \(B\) is the lower limit topology on \(\mathbb R\) if
—>> \(\mathbb B’ = {[a,b) : a,b\epsilon\mathbb R; a<b}\)
\(\mathbb B’ = {(a,b] : a,b\epsilon\mathbb R; a<b}\)
\(\mathbb B’ = {[a,b] : a,b\epsilon\mathbb R; a<b}\)
\(\mathbb B’ = (a,b) : a,b\epsilon\mathbb R; a<b\)
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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