When one suddenly starts studying compactness and connectedness and other topological concepts in college, one is likely to get confused.
Where did all these concepts come from? Then, seemingly intuitive properties of start being proven using these alien notions. Forming a big picture which includes these concepts seems difficult. One still thinks about the real number line in intuitive terms. As for these terms, although one eventually begins to understand them, acceptance of such notions as tools for studying is difficult.
This is an attempt to mend that divide.
Connectedness, compactness, boundedness, and other seemingly arbitrary topological notions are properties of (along with other topological spaces) that we do not come across before they’re thrust on us. Hence the difficulty in accepting them in situations that are entirely too familiar and plain to us. If the situations too were alien, accepting such tools in analyzing them would perhaps be easier.
These properties are important, as lots of them are conserved in continuous mappings. Hence, whether one space can be continuously mapped to another can be exhaustively determined using these tools. I repeat, nothing about these tools is difficult except for the fact that they’re new to us, and the fact that they’re new to us has not been explicitly stated.
And there is nothing holy about these tools. I hypothesize that many more such properties which are conserved in continuous mappings are still to be discovered. And when they are discovered, we will be able to answer questions that stand unanswered today.
Topology has much in common with Physics. We’re looking around, searching for the laws of nature, but somehow making do with the laws we currently know.