The theorem states that the continuous image of a compact set is compact. Compactness is thus a property that is preserved by continuous transformation; compactness is, in other words, a topological property. The notes on continuity and connectedness establish that connectedness is also a topological property.

Does continuity preserve hausdorff?

By elementary theorems a continuous function is always preserving. McMillan [Pacific J. ... 32 (1970) 479] proved in 1970 that if X is Hausdorff, locally connected and Frechét, Y is Hausdorff, then the converse is also true: any preserving function is continuous.

What is continuity in a topological space?

Definition A function f:X → Y from a topological space X to a topological space Y is said to be continuous if f−1(V ) is an open set in X for every open set V in Y , where f−1(V ) ≡ {x ∈ X : f(x) ∈ V }. ... The function f is continuous if and only if f−1(G) is closed in X for every closed subset G of Y .