Past exam of the options I course of the University of Oxford / 2008 / E / E1 / a

Table of contents
- i a
  - Solution i
- ii a
  - Solution ii

i

Solution

A map of topological spaces is continuous when the preimage

f^{- 1} (U)

is open in

X

for every open set

U

of

Y

. Equivalently, the preimage of every closed subset of

Y

is closed in

X

.

ii

Solution

For every

ε > 0

there must be a

δ > 0

such that

dist_{X} (x, x_{0}) < δ ⟹ dist_{Y} (f (x), f (x_{0})) < ε .

The choice of

δ

may depend on

x_{0}

and

ε

. This is the epsilon-delta definition of continuity.

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