This paper deals with fixed point theory and fixed point property in minimal spaces. We will prove that under some conditions f : (X,M) → (X,M) has a fixed point if and only if for each m-open cover {Bα} for X there is at least one x ∈ X such that both x and f(x) belong to a common Bα. Further, it is shown that if (X,M) has the fixed point property, then its minimal retract subset enjoys this property.