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In my opinion, I think it's okay that you treat compactness as a clever trick as you said. But as it is an abstract concept, I think it's recommended to take the concept as itself, not comparing to real examples.

Maybe the concept of compactness is motivated from Heine-Borel property, but even without a motivation, I think the definition of compactness itself is very natural.

There are many branches of mathematics and each branch has its main vocabularies. For example, in number theory, one's main vocabulary is an "integer". Just like this, "openness" is a main vocabulary in general topology. (The question of asking why openness is a main vocabulary is a completely different question and this is well-answered in wikipedia.) So when you do general topology, you play with open sets. And definitely it's hard to make an argument on infinite open objects, but compactness ease this so that when you play with compactness you can only consider finite objects (for each compact set).

If you take concepts in this way, definitions of Lindelöf space and paracompact space and etc are very natural just like compactness. I'm quite sure no one has a clear "real life example" of paracompactness but almost all of people here think the definition is very natural.

In my opinion, I think it's okay that you treat compactness as a clever trick as you said. But as it is an abstract concept, I think it's recommended to take the concept as itself, not comparing to real examples.

Maybe the concept of compactness is motivated from Heine-Borel property, but even without a motivation, I think the definition of compactness itself is very natural.

There are many branches of mathematics and each branch has its main vocabularies. For example, in number theory, one's main vocabulary is an "integer". Just like this, "openness" is a main vocabulary in general topology. (The question of asking why openness is a main vocabulary is a completely different question and this is well-answered in Wikipedia.) So when you do general topology, you play with open sets. And definitely it's hard to make an argument on infinite open objects, but compactness ease this so that when you play with compactness you can only consider finite objects (for each compact set).

If you take concepts in this way, definitions of Lindelöf space and paracompact space and etc are very natural just like compactness. I'm quite sure no one has a clear "real life example" of paracompactness but almost all of people here think the definition is very natural.

In my opinion, I think it's okay that you treat compactness as a clever trick as you said. But as it is an abstract concept, I think it's recommended to take the concept as itself, not comparing to real examples.

Maybe the concept of compactness is motivated from Heine-Borel property, but even without a motivation, I think the definition of compactness itself is very natural.

There are many branches of mathematics and each branch has its main vocabularies. For example, in number theory, one's main vocabulary is an "integer". Just like this, "openess" is a main vocabulary in general topology. (The question of asking why openess is a main vocabulary is a completely different question and this is well-answered in wikipedia.) So when you do general topology, you play with open sets. And definitely it's hard to make an argument on infinite open objects, but compactness ease this so that when you play with compactness you can only consider finite objects (for each compact set).

If you take concepts in this way, definitions of Lindelöf space and paracompact space and etc are very natural just like compactness. I'm quite sure no one has a clear "real life example" of paracompactness but almost all of people here think the definition is very natural.

In my opinion, I think it's okay that you treat compactness as a clever trick as you said. But as it is an abstract concept, I think it's recommended to take the concept as itself, not comparing to real examples.

Maybe the concept of compactness is motivated from Heine-Borel property, but even without a motivation, I think the definition of compactness itself is very natural.

There are many branches of mathematics and each branch has its main vocabularies. For example, in number theory, one's main vocabulary is an "integer". Just like this, "openness" is a main vocabulary in general topology. (The question of asking why openness is a main vocabulary is a completely different question and this is well-answered in wikipedia.) So when you do general topology, you play with open sets. And definitely it's hard to make an argument on infinite open objects, but compactness ease this so that when you play with compactness you can only consider finite objects (for each compact set).

If you take concepts in this way, definitions of Lindelöf space and paracompact space and etc are very natural just like compactness. I'm quite sure no one has a clear "real life example" of paracompactness but almost all of people here think the definition is very natural.

In my opinion, I think it's okay that you treat compactness as a clever trick as you said. But as it is an abstract concept, I think it's recommended to take the concept as itself, not comparing to real examples.

Maybe the concept of compantness is motivated from Heine-Borel property, but even without a motivation, I think the definition of compactness itself is very natural.

There are many branches of mathematics and each branch has its main vocabularies. For example, in number theory, one's main vocabulary is an "integer". Just like this, "openess" is a main vocabulary in general topology. (The question of asking why openess is a main vocabulary is a completely different question and this is well-answered in wikipedia.) So when you do general topology, you play with open sets. And definitely it's hard to make an argument on infinite open objects, but compactness ease this so that when you play with compactness you can only consider finite objects (for each compact set).

If you take concepts in this way, definitions of Lindelöf space and paracompact space and etc are very natural just like compactness. I'm quite sure no one has a clear "real life example" of paracompactness but almost all of people here think the definition is very natural.

In my opinion, I think it's okay that you treat compactness as a clever trick as you said. But as it is an abstract concept, I think it's recommended to take the concept as itself, not comparing to real examples.

Maybe the concept of compactness is motivated from Heine-Borel property, but even without a motivation, I think the definition of compactness itself is very natural.

There are many branches of mathematics and each branch has its main vocabularies. For example, in number theory, one's main vocabulary is an "integer". Just like this, "openess" is a main vocabulary in general topology. (The question of asking why openess is a main vocabulary is a completely different question and this is well-answered in wikipedia.) So when you do general topology, you play with open sets. And definitely it's hard to make an argument on infinite open objects, but compactness ease this so that when you play with compactness you can only consider finite objects (for each compact set).

If you take concepts in this way, definitions of Lindelöf space and paracompact space and etc are very natural just like compactness. I'm quite sure no one has a clear "real life example" of paracompactness but almost all of people here think the definition is very natural.