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When we delve into the realm of topology, we encounter the fundamental concept of a topological space. This is a set, let's call it X, paired with a collection of subsets deemed open sets. These sets adhere to specific rules; namely, the complete set and the empty set are within this collection, any union of open sets is open, and the intersection of finitely many open sets is also open.

These open sets outline the structure of X and allow us to discuss continuity, convergence, and connectedness within this framework. Simplistically, a topological space can be imagined as a playground where points can 'move' and 'cluster' according to the rules established by the open sets, creating an abstract stage for exploring geometric and analytical concepts without relying on notions like distance.

In topology, a function is considered a continuous surjection if every open set's preimage is open in the function's domain and the function hits every point in the target space at least once. This means that continuity preserves the openness of sets under the function's action, and being surjective ensures that the whole of the target space is covered.

Picture a painter with a canvas, the continuous surjective function is like a brush that paints across the entire canvas without leaving gaps and does so smoothly, without any abrupt changes. That imagery gives us a glimpse at how topology views functions – they're judged based on how they line up open sets between spaces rather than focusing on their algebraic properties alone.

Consider the closure of a set like the boundary that embraces all its points and also reaches out to lightly grasp any limits of sequences within that set. Formally, the closure of a set A in a topological space, denoted as \(\bar{A}\), is the smallest closed set that contains A. This closed set includes not only A itself but any points that are so close to A, they might as well be considered a part of A mathematically.

The closure of a set intertwines with the concept of connectedness. If A is a connected set, its closure will not have any 'holes' or 'breaks', preserving the connection across its entire extent. This continuity of connection allows us to see the closure as an extension of the original set, padding it out with all its possible limits to ensure no breakage in its internal structure.

A topological space can have various subsets that we can investigate to reveal the space's nature. When we find disjoint non-empty open sets, we uncover sets that are both open and do not share any points between them. An open set can be thought of as one that does not contain its boundary, and being disjoint is akin to saying that these sets are completely separate entities with no overlap.

When we talk about connected sets, we imply that the set cannot be broken into disjoint non-empty open sets. This means the set is, in a sense, one whole piece without any internal separation. If we could break a connected set into such disjoint open subsets, it would be like finding a fracture in a previously thought solid object - it would defy the very essence of being a connected set.