Let Cb(X) be the C∗-algebra of bounded continuous functions on some non-compact, but locally compact Hausdorff space X. Moreover, let $\mathfrak {A}_{0}$ be some ideal and $\mathfrak {A}_{1}$ be some unital C∗-subalgebra of Cb(X). For $\mathfrak {A}_{0}$ and $\mathfrak {A}_{1}$ having trivial intersection, we show that the spectrum of their vector space sum equals the disjoint union of their individual spectra, whereas their topologies are nontrivially interwoven. Indeed, they form a so-called twisted-sum topology which we will investigate beforehand. Within the whole framework, e.g., the one-point compactification of X and the spectrum of the algebra of asymptotically almost periodic functions can be described.