In R with its usual metric, how many limit points does A={-4,1,7} have?

Let X={a,b,c} have the discrete metric: distinct points are distance 1 apart. Which statement about {a} is true?

• It is open in X but is not closed in X.
• It is both open and closed in X.
• It is closed in X but is not open in X.
• It is neither open nor closed in X.

In R with its usual metric, what is the interior of [0,1]?

• The closed interval [0,1].
• The open interval (0,1).
• The set containing only 0 and 1.

In every metric space, arbitrary unions and finite intersections of open sets are open. An infinite intersection need not be open.

In R with its usual metric, let A=(-2,0) union (0,3). How many points lie in the interior of the closure of A but not in A?

For a map f:X->Y between metric spaces, which condition is equivalent to continuity?

• The image of every open set in X is open in Y.
• The preimage of every singleton in Y contains exactly one point of X.
• The preimage of every open set in Y is open in X.

A proof takes x in a union of open sets and tries to choose a radius that works in every member. Which change gives a valid proof?

• Use a ball from one member containing x; it also lies in the union.
• Take the minimum radius over all members, including those not containing x.
• Require x to belong to every member before choosing a ball in the union.
• Use the largest member radius, even if its ball leaves the entire union.

In any metric space, the intersection of an arbitrary family of closed sets is closed.

Let f:R->R be 0 for x<=0 and 1 for x>0. For V=(-0.5,0.5), which statement about f^-1(V) witnesses discontinuity?

• It is (0,infinity); no positive input has a ball contained in this set.
• It is all of R; both constant pieces make f continuous at 0.
• It is (-infinity,0]; no ball about 0 lies inside this set.

For f:X->Y between metric spaces, assume every open target set has an open preimage. Fix a in X and epsilon>0. Which construction proves continuity at a?

• Take a target ball B_Y(f(a),epsilon); use epsilon itself as the source radius without examining any preimage.
• Take f(B_X(a,epsilon)); call this image open and use a target ball inside it to choose the source radius.
• Take G=f^-1(B_Y(f(a),epsilon)); openness at a gives delta>0 with B_X(a,delta) inside G.
• Take G=f^-1({f(a)}); claim that this preimage must be open and contains a source ball for every epsilon.