<p>We introduce and explore two generalized notions of connectedness in topological spaces, extending the classical concept of connectedness. A space is said to be <i>connected at infinity</i> if any “small” subset can be removed by another “small” subset, leaving the remaining space connected. The notion of an <i>ideal</i>, understood as a collection of “small” subsets, naturally arises in this context. More formally, a space is called <i>connected modulo an ideal of subsets</i> if every element of the ideal is contained within another element of the ideal whose complement is connected. Next, we define a space to be <i>boundary connected modulo an ideal of its subsets</i> if it does not contain a “large” subset whose complement is also “large” and whose boundary is “small.” We explore the relationship between these new notions and previously established concepts of connectedness, demonstrating that many standard results on connectedness have counterparts within these generalized frameworks. Our approach expands the understanding of connectedness in topology and lays the foundation for further research into the interplay between ideals and topological properties.</p>