Let X be a completely regular topological space. We study closed ideals H of $C_B(X)$, the normed algebra of bounded continuous scalar-valued mappings on X equipped with pointwise addition and multiplication and the supremum norm, which are non-vanishing, in the sense that, there is no point of X at which every element of H vanishes. This is done by studying the (unique) locally compact Hausdorff space Y associated to H in such a way that H and $C_0(Y)$ are isometrically isomorphic. We are interested in various connectedness properties of Y. In particular, we present necessary and sufficient (algebraic) conditions for H such that Y satisfies (topological) properties such as locally connectedness, total disconnectedness, zero-dimensionality, strong zero-dimensionality, total separatedness or extremal disconnectedness.

令X为完全规则的拓扑空间。我们研究H的封闭理想H$C_{乙}（X）$，X上的无界连续标量值映射的范式代数具有逐点加法和乘法以及至高范数，它们是不存在的，从某种意义上说，X的任何点都不会消失H的每个元素。这是通过研究与H关联的（唯一）局部紧凑的Hausdorff空间Y来完成的，使得H和$C_0（ÿ）$是等轴同构的。我们对Y的各种连接性质感兴趣。尤其是，我们为H提供了必要且充分的（代数）条件，以使Y满足（拓扑）性质，例如局部连通性，总断开性，零维，强零维，总分离性或极端断开性。