MTH402
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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