Hint  Answer 
The closure of A intersecting the closure of the compliment of A.  
A space X is said to be this if for every neighborhood U of x, there is a connected neighborhood V of x contained in U.  
All closed sets containing A.  
Take a basis for a topology on a product space Pi (Xa) the collection of all sets of the of the form Pi(Ua) with Ua is open in Xa. The topology generated by this is called a?  
When the set X has a topology on it, it is called a what?  
A space X is said to be this if every pair of points of X can be joined by a path in X.  
If both the function f and the inverse function are continuous, the f is called a what?  
A subset of A of a topological space X is said to be what if the set X  A is open.  
This of X is a pair U, V of disjoint nonempty open subsets of X whose union is X.  
This is the sup(d(a1, a2) for all a1 and a2)  
If X is a space and A is a set and if p : X> A is surjective, then there exists one topology T on A relative to p which is a quotient map, making the topology a what?  
A topological space X is called this if for each pair of distinct points of X, there exist neighborhoods U1 and U2 of x1 and x2 respectively that are disjoint.  
p is said to be this provided a subset U of Y is open in Y if and only if p1(a) is closed in X.  
If X is a topological space with topology T, we say that a subset U of X is a what if U belongs to T?  
If T'>T, then T is what?  
A subset C of X is this if C contains every set p1(y) that it intersects  
sup{d(xa,ya)}  
The point x is a limit point of A if and every neighborhood of x contains infinitely many points of A.  
If X is pathconnected at each of its points then it is said to be this.  
The functions f1: A >X and f2 A > Y which are continuous are called?  
Given a subset A of a topological space X, the what of A is defined as the union of all open sets contained in A.  
If T'>T, then T' is what?  
In the quotient topology induced by p, the space X* is called this.  
 Hint  Answer 
max{x1y1 ... xnyn}  
S for a topology on X is a collection of subsets of X where the union equals X.  
d(x,y) >= 0 if x=/= y, d(x,y) = d(y,x), and the triangle inequality holds.  
This is the collection T of all unions of finite intersections of elements of S.  
Any property of X that is entirely expressed in terms of the topology of X yields, via the correspondence f, the corresponding property for the space Y.  
The null set and the empty set are in it, it contains the unions of elements of the subcollection, and arbitrary intersections.  
x is this if every neighborhood of x intersects A in some point other than x itself.  
A space X is said to be this if for every neighborhood U of x, there is a pathconnected neighborhood V of x contained in U.  
This on X x Y is the topology having as basis the collection B of all sets of the form U x V, where U is an open subset of X and V is an open subset of Y.  
If X is a topological space, X is said to be what if there exists a metric d on the set X that induces the topology of X.  
The space X is said to be what if there does not exist a separation of X.  
This is a metrizable space X together with a specific metric d that gives the topology of X.  
B is what is for each x in X there is at least one B containing x and if x belongs to B1 and B2, then B3 contains x and B3=B1^B2.  
The equivalence classes are these.  
An open set containing x.  
A function F:X>Y is said to be what for each open subset V or Y, the set f1 (V) is an open subset of X.  
If B is the collection of (a,b) of X, [a0, b) of X and (a,b0] of X, then B is the basis for what?  
A subset U of X is said to be open in X if for each x in U, there is a B in B' such that x is in B and B is a subset of U.  
Let pi1: XxY > X be defined by the equation pi1(x,y) =x and pi2: XxY > Y defined by pi2(x,y)=y, then pi1 and pi2 are called what on the first and second factors of XxY.  
If X is connected at each of its points, it is this.  
Let X be a topological space with topology T. If Y is a subset of X, the collection Y^U, U in T is a topology on Y called a what?  
A map f: X> Y is said to be a what if for every open set U of X, the set f(U) is open in Y.  
[(x1y1)^2 + ... + (xnyn)^2]^1/2  

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