4 Homeomorphisms

Question 1

Subspace topology (relative topology, induced topology)

Answer 1

Let Y be a non-empty subset of a topological space (X, T). The collection T_Y = {O n Y: O e T} of subsets of Y is a topology on Y called the subspace topology (or the relative topology or the induced topology or the topology induced on Y by T). The topological space (Y, T_Y) is said to a subspace of (X, T).

Question 2

Homeomorphic, homeomorphism

Answer 2

Let (X, T) and (Y, T1) be topological spaces. Then they are said to be homeomorphic if there exists a function f: X -> Y which has the following properties:
i) f is injective
ii) f is surjective
iii) for each U e T1, f-1(U) e T, and
iv) for each V e T, f(V) e T1
Further, the map f is said to be a homeomorphism between (X, T) and (Y, T1). We write (X, T) =~ (Y, T1).

Question 3

Interval

Answer 3

A subset S of R is said to be an interval if it has the following property: if x e S, z e S, and y e R are such that x < y < z, then y e S.