Comment. Point set topology quiz
Comment. This quiz is designed to test your knowledge of point set topology notions in metric spaces,
Comment. such as open and closed sets, compact and connected sets, interior and adherent points, etc.
Shuffle Questions.
Shuffle Answers.
Question. Let E be a subset of a metric space (X,d). What does it mean for x to be an adherent point of E?
Correct Answer. For every epsilon > 0, there exists a y in E such that d(y,x) < epsilon.
Answer. For every epsilon > 0, there exists a y in E such that 0 < d(y,x) < epsilon.
Comment. This is what it means for x to be a limit point of E, not an adherent point.
Answer. There exists epsilon > 0 and y in E such that 0 < d(y,x) < epsilon.
Answer. For every epsilon > 0 and all y in E, we have d(y,x) < epsilon.
Answer. For every y in E, there exists epsilon > 0 such that d(y,x) < epsilon.
Answer. There exists epsilon > 0 such that 0 < d(y,x) < epsilon for all y in E.
Answer. There exists y in E such that d(y,x) < epsilon for all epsilon > 0.
Question. Let E be a subset of a metric space (X,d). What does it mean for x to NOT be an adherent point of E?
Correct Answer. There exists an epsilon > 0 such that d(y,x) >= epsilon for all y in E.
Answer. There exists an epsilon > 0 and y in E such that d(y,x) >= epsilon.
Answer. For every epsilon > 0 there exists y in E such that d(y,x) >= epsilon.
Answer. For every epsilon > 0 we have d(y,x) >= epsilon for every y in E.
Answer. For every y in E there exists an epsilon > 0 such that d(y,x) >= epsilon.
Answer. There exists y in E such that d(y,x) >= epsilon for every epsilon > 0.
Answer. There exists an epsilon > 0 such that y in E whenever d(y,x) >= epsilon.
Question. Let E be a subset of a metric space (X,d). What does it mean for x to be an adherent point of E?
Correct Answer. There exists a sequence x_1, x_2, x_3, ... in E which converges to x.
Answer. There exists a sequence x_1, x_2, x_3, ... in E-{x} which converges to x.
Comment. This is what it means for x to be a limit point of E, not an adherent point.
Answer. Every sequence x_1, x_2, x_3, ... in E which converges, converges to x.
Answer. Every sequence x_1, x_2, x_3, ... in E-{x} converges to x.
Answer. Every Cauchy sequence x_1, x_2, x_3, ... in E converges to x.
Answer. There exists a sequence x_1, x_2, x_3, ... in E-{x} which is a Cauchy sequence.
Answer. Every sequence x_1, x_2, x_3, ... which converges to x, must lie in X.
Question. Let E be a subset of a metric space (X, d_X), let f: X -> Y be a function from X to another metric space (Y,d_Y), let x be an adherent point of E, and let L be a point in Y. What does it mean for lim_{y -> x; y in E} f(x) to equal L?
Answer. For every epsilon > 0, there exists a delta > 0 such that d_Y(f(x'),L) < delta for all x' in E for which d_X(x',x) < epsilon.
Answer. For every epsilon > 0, there exists a delta > 0 such that d_X(x',x) < epsilon for all x' in E for which d_Y(f(x'),L) < delta.
Correct Answer. For every epsilon > 0, there exists a delta > 0 such that d_Y(f(x'),L) < epsilon for all x' in E for which d_X(x',x) < delta.
Answer. For every epsilon > 0 and delta > 0, we have d_Y(f(x'),L) < epsilon for all x' in E for which d_X(x',x) < delta.
Answer. For every epsilon > 0 and every x' in E, there exists delta > 0 such that d_Y(f(x'),L) < epsilon if d_X(x',x) < delta.
Answer. For every epsilon > 0, there exists a delta > 0 such that d_Y(f(x'),L) < epsilon and d_X(x',x) < delta for some x' in E.
Answer. For every epsilon > 0, there exists a delta > 0 such that d_Y(f(x'),L) < epsilon and d_X(x',x) < delta for all x' in E.
Question. Let X be a subset of R, let f: X -> R be a function, and let x be an element of X. What does it mean for f to be continuous at x?
Answer. For every epsilon > 0, there exists a delta > 0 such that d_X(f(x'),f(x)) < delta for all x' in X for which d_X(x',x) < epsilon.
Answer. For every epsilon > 0, there exists a delta > 0 such that d_X(x',x) < epsilon for all x' in X for which d_Y(f(x'),f(x)) < delta.
Correct Answer. For every epsilon > 0, there exists a delta > 0 such that d_Y(f(x'),f(x)) < epsilon for all x' in X for which d_X(x',x) < delta.
Answer. For every epsilon > 0 and delta > 0, we have d_Y(f(x'),f(x)) < epsilon for all x' in X for which d_X(x',x) < delta.
Answer. For every epsilon > 0 and every x' in X, there exists delta > 0 such that d_Y(f(x'),f(x)) < epsilon if d_X(x',x) < delta.
Answer. For every epsilon > 0, there exists a delta > 0 such that d_Y(f(x'),f(x)) < epsilon and d_X(x',x) < delta for some x' in X.
Answer. For every epsilon > 0, there exists a delta > 0 such that d_Y(f(x'),f(x)) < epsilon and d_X(x',x) < delta for all x' in X.
Question. Let E be a subset of a metric space (X,d_X), let f: X -> Y be a function from X to another metric space (Y,d_Y), and let x be an adherent point of E. What does it mean for lim_{y -> x; y in X} f(x) to equal L?
Correct Answer. For every sequence x_1, x_2, x_3, ... in E which converges to x, the sequence f(x_1), f(x_2), f(x_3), ... converges to L.
Answer. There exists a sequence x_1, x_2, x_3, ... in E converging to x, such that f(x_1), f(x_2), f(x_3), ... converges to L.
Answer. Whenever f(x_1), f(x_2), ... converges to L, the sequence x_1, x_2, x_3, ... , must then converge to x.
Answer. For every sequence x_1, x_2, x_3, ... in E which converges to L, the sequence f(x_1), f(x_2), f(x_3), ... converges to x.
Answer. Whenever f(x_1), f(x_2), ... converges to L, the sequence x_1, x_2, x_3, ... , must lie in E and converge to x.
Answer. Whenever x_1, x_2, ... converges to x, the sequence f(x_1), f(x_2), f(x_3), ... , must lie in E and converge to L.
Answer. For every sequence x_1, x_2, x_3, ... in E which converges to L, the sequence f(x_1), f(x_2), f(x_3), ... converges to x.
Question. Let E be a subset of a metric space (X, d_X), let f: X -> Y be a function from X to another metric space (Y,d), and let x be an element of E. What does it mean for f to be continuous at x?
Correct Answer. For every sequence x_1, x_2, x_3, ... in E which converges to x, the sequence f(x_1), f(x_2), f(x_3), ... converges to f(x).
Answer. There exists a sequence x_1, x_2, x_3, ... in E converging to x, such that f(x_1), f(x_2), f(x_3), ... converges to f(x).
Answer. Whenever f(x_1), f(x_2), ... converges to f(x), the sequence x_1, x_2, x_3, ... , must then converge to x.
Answer. Every sequence f(x_1), f(x_2), ... which is convergent, converges to f(x).
Answer. Whenever f(x_1), f(x_2), ... converges to f(x), the sequence x_1, x_2, x_3, ... , must lie in E and converge to x.
Answer. Whenever x_1, x_2, ... converges to x, the sequence f(x_1), f(x_2), f(x_3), ... , must lie in E and converge to f(x).
Answer. For every sequence x_1, x_2, x_3, ... in E, the sequence f(x_1), f(x_2), f(x_3), ... converges to f(x).
Question. Let f: X -> Y be a function from a metric space (X,d_X) to a metric space (Y, d_Y). What does it mean for f to be continuous on X?
Correct Answer. For every epsilon > 0 and x in X, there exists a delta > 0 such that d_Y(f(x'),f(x)) < epsilon for all x' in X for which d_X(x',x) < delta.
Answer. For every epsilon > 0, there exists a delta > 0 such that d_Y(f(x'),f(x)) < epsilon for all x, x' in X for which d_X(x',x) < delta.
Comment. This is what it means for f to be uniformly continuous on X.
Answer. For every epsilon > 0, there exists a delta > 0 and x,x' in X such that d_Y(f(x'),f(x)) < delta and d_X(x',x) < epsilon.
Answer. For every epsilon > 0 and x in X, there exists a delta > 0 such that d_Y(f(x'),f(x)) < delta for all x' in X for which d_X(x',x) < epsilon.
Answer. For every x,x' in X, there exists an epsilon > 0 and delta > 0 such that d_Y(f(x'),f(x)) < delta and d_X(x',x) < epsilon.
Answer. For every epsilon > 0 and x in X, there exists a delta > 0 such that d_X(x',x) < epsilon for all x' in X for which d_Y(f(x'),f(x)) < delta.
Answer. For every epsilon > 0, there exists a delta > 0 such that d_X(x',x) < epsilon for all x,x' in X for which d_Y(f(x'),f(x)) < delta.
Question. Let f: X -> Y be a function from a metric space (X,d_X) to a metric space (Y, d_Y). What does it mean for f to be uniformly continuous on X?
Answer. For every epsilon > 0 and x in X, there exists a delta > 0 such that d_Y(f(x'),f(x)) < epsilon for all x' in X for which d_X(x',x) < delta.
Comment. This is what it means for f to be continuous on X.
Correct Answer. For every epsilon > 0, there exists a delta > 0 such that d_Y(f(x'),f(x)) < epsilon for all x, x' in X for which d_X(x',x) < delta.
Answer. For every epsilon > 0, there exists a delta > 0 and x,x' in X such that d_Y(f(x'),f(x)) < delta and d_X(x',x) < epsilon.
Answer. For every epsilon > 0 and x in X, there exists a delta > 0 such that d_Y(f(x'),f(x)) < delta for all x' in X for which d_X(x',x) < epsilon.
Answer. For every x,x' in X, there exists an epsilon > 0 and delta > 0 such that d_Y(f(x'),f(x)) < delta and d_X(x',x) < epsilon.
Answer. For every epsilon > 0 and x in X, there exists a delta > 0 such that d_X(x',x) < epsilon for all x' in X for which d_Y(f(x'),f(x)) < delta.
Answer. For every epsilon > 0, there exists a delta > 0 such that d_X(x',x) < epsilon for all x,x' in X for which d_Y(f(x'),f(x)) < delta.
Question. Let E be a subset of a metric space (X,d). If E is both open and closed, then we can conclude that
Correct Answer. The boundary of E is the empty set.
Answer. E is either the empty set, or the whole space X.
Comment. This is a sufficient condition for E to be both open and closed, but not necessary (unless X is connected).
Answer. E is the empty set.
Answer. X is the empty set.
Answer. X is disconnected.
Comment. This is only true if E is a non-empty proper subset of X.
Answer. E is disconnected.
Answer. E has no interior and no exterior.
Question. Let E be a subset of a metric space (X,d). We say that E is complete if and only if
Correct Answer. Every Cauchy sequence in E, converges in E.
Answer. Every Cauchy sequence in E, converges in X.
Answer. Every convergent sequence in E is Cauchy.
Answer. Every sequence in E has a subsequence which converges in E.
Comment. This is what it means for E to be compact, not complete.
Answer. Every sequence in E has a subsequence which converges in X.
Answer. Every Cauchy sequence in E has a convergent subsequence.
Answer. Every sequence in E has a Cauchy subsequence.
Question. Let E be a subset of a metric space (X,d). We say that E is compact if and only if
Answer. Every Cauchy sequence in E, converges in E.
Comment. This is what it means for E to be complete, not compact.
Answer. Every Cauchy sequence in E, converges in X.
Answer. Every convergent sequence in E is Cauchy.
Correct Answer. Every sequence in E has a subsequence which converges in E.
Answer. Every sequence in E has a subsequence which converges in X.
Answer. Every Cauchy sequence in E has a convergent subsequence.
Answer. Every sequence in E has a Cauchy subsequence.
Question. Let E be a subset of a metric space (X,d). We say that E is open if and only if
Correct Answer. Every point in E is an interior point.
Answer. E contains all of its interior points.
Answer. E contains all of its adherent points.
Comment. This is what it means for E to be closed.
Answer. Every point in E is an adherent point.
Answer. E has no boundary.
Answer. E has no exterior.
Answer. E does not contain any adherent points.
Question. Let E be a subset of a metric space (X,d). We say that E is closed if and only if
Answer. Every point in E is an interior point.
Comment. This is what it means for E to be open.
Answer. E contains all of its interior points.
Correct Answer. E contains all of its adherent points.
Answer. Every point in E is an adherent point.
Answer. E has no boundary.
Answer. E has no interior.
Answer. E does not contain any interior points.
Question. Let E be a subset of a metric space (X,d). We say that E is disconnected if and only if
Correct Answer. There exist disjoint open sets V, W in X which cover E and which both have non-empty intersection with E.
Answer. There exist disjoint non-empty open sets V, W in X which cover E.
Answer. There exist disjoint open sets V, W in X which both have non-empty intersection with E.
Answer. There exist disjoint non-empty open sets V, W in X whose union is E.
Answer. There exist open sets V, W in X whose union contains E, and who both have non-empty intersection with E.
Answer. Both E and X\E are open and non-empty.
Answer. E can be partitioned into two disjoint non-empty pieces, one of which is open and the other of which is closed.
Question. In the real line R with the usual metric, the rationals Q are
Don't Shuffle These Answers.
Answer. Bounded.
Answer. Closed.
Answer. Compact.
Answer. Complete.
Answer. Connected.
Answer. Open.
Correct Answer. None of the above.
Question. In the real line R with the _discrete_ metric, the rationals Q are
Correct Answer. Bounded, Closed, Complete, and Open.
Answer. Closed, Compact, Complete and Connected.
Answer. Open, Bounded, Connected, and Complete.
Answer. Closed, Bounded, Complete, and Compact.
Answer. Complete, Connected, Open, and Bounded.
Answer. Open, Closed, Bounded, and Compact.
Answer. None of the above.