1 Topological spaces

Question 1

A topology / topological space

Answer 1

Let X be a non-empty set. A set T of subsets of X is said to be a topology on X if:
1) X and the empty set O\ belong to T
2) The union of any number of sets in T belongs to T
3) The intersection of any two sets of in T belongs to T.
The pair (X, T) is called a topological space.

Question 2

Discrete topology / Discrete space

Answer 2

Let X be any non-empty set and let T be the collection of all subsets of X. Then T is called the discrete topology on the set X. The topological space (X, T) is called a discrete space.

Question 3

Indiscrete topology / indiscrete space

Answer 3

Let X be any non-empty set and T = (X, O/). Then T is called the indiscrete topology and (X, T) is said to be an indiscrete space.

Question 4

Open sets

Answer 4

Let (X, T) be any topological space. Then the members of T are said to be open sets.

Question 5

Closed set

Answer 5

Let (X, T) be a topological space. A subset S of X is said to be a closed set in (X, T) if its complement in X, namely X \ S, is open in (X, T).

Question 6

Clopen set

Answer 6

A subset S of a topological space (X, T) is said to be clopen if its both open and closed in (X,T).

Question 7

Finite-closed topology (cofinite topology)

Answer 7

Let X be any non-empty set. A topology T on X is called the finite-closed topology or the cofinite topology if the closed subsets of X are X and all finite subsets of X; that is, the open sets are O/ and all subset of X which have finite complements.

Question 8

Injective function

Answer 8

Let f be a function from a set X into a set Y.

The function f is said to be one-to-one or injective if f(x1) = f(x2) implies x1 = x2.

Question 9

Surjective function

Answer 9

A function f from set X into a set Y is said to be surjective or onto if for each y in Y there exists an x in X such that f(x) = y

Question 10

Bijective function

Answer 10

A function f is said to be bijective if its both injective and surjective.

Question 11

Inverse function

Answer 11

Let f be a function from a set X into a set Y. The function f is said to have an inverse if there exists a function g of Y into X such that g(f(x)) = x for all x and f(g(y)) = y for all y. The function g is then called an inverse function of f.

Question 12

Inverse image

Answer 12

Let f be a function from a set X into a set Y. If S is any subset of Y, then the set f-1(S) is defined by
f-1(S) = {x: x e X and f(x) e S}.
The subset f-1(S) is said to be the inverse image of S.