MTH402
Question: If /( {x_n } /) is a sequence of elements of a topological space X. If for all neighbourhoods U of /(x∈X, /) , there exists N in the set of natural numbers such that for all /( n≥N,x_(n )∈U /) is said to be
Answer: converges to x.
Question: If A is a subset of a topological space X, then the intersection of closed sets containing A is called
Answer: the closure of A
Question: If a topological space X is such that for all /( x,y∈X with x ≠y /) there exists /( U_(x )∈ N(x) such that yɇ U_x. /) Then X is said to satisfy
Answer: the first separation axiom /( T_1/)
Question: Let X be a topological space. If for any closed set F of X and any point /( x∈X \\F,/) , there exist open sets /( U_x,U_(y )∈X /) such that /(〖x ∈ U〗_x,F subset of U_(Y ) and U_x∩ U_F=∅ ,/) , then
Answer: X is regular.
Question: If X is a non-empty set, /( Ï„ /) a collection of subsets of X. then the pair /( X,Ï„ /) is called a topological space if the following conditions are satisfied except:
Answer: Every sequence in X converges to a point in X
Question: Assume that A is a subset of a topological space X, then the union of all open sets contained in A is called
Answer: the interior of A
Question: Let X be a topological space. If for all /( x,y∈X with x≠y , /) there exist /(U_x∈N(x) ,U_(y )∈N(y) /) such that /( U_xU_y=∅./) . Then X is called
Answer: a Hausdorff space
Question: Let X and Y be nonempty sets. If for all /( y∈Y /) there exists a unique /( x∈X /) such that /( f(x)=y, /) then f is called
Answer: injective
Question: Let X and Y be topological spaces. If a function /(f:X→Y /) such that for each open subset /( U_y /) of Y, the set /( f^(-1) (U_y )/) is an open subset of X, where /(f^(-1) (U_y )= {x∈X:f(x)∈ U_y },/) then f is said to be
Answer: continuous
Question: If a subset A of a topological space X is such that it contains all its limit points, then A is
Answer: closed
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