What is the meaning of phrase,"Compactness and Connectedness are intrinsic properties of a topological space"?
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That compactness and connectedness are ESSENTIAL (or fundamental). Those properties can’t miss in a Topological Space! – James Garrett Oct 26 '17 at 22:41
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2@AdriánNaranjo:Does properties can’t miss in a Topological Space!?? – Styles Oct 26 '17 at 22:43
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1I fixed the typo! Sorry – James Garrett Oct 26 '17 at 22:44
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2@AdriánNaranjo Don't see the fix. "can't miss in a Topological Space!" is meaningless. – Thomas Andrews Oct 26 '17 at 22:45
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2Can you provide some context? – Rob Arthan Oct 26 '17 at 22:46
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@Rob Arthan:I'm watching a lecture on youtube on complex analysis,there the professor is quoting this phrase – Styles Oct 26 '17 at 22:48
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1Related. – Pedro Oct 26 '17 at 22:51
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This means that these properties are preserved by homeomorphism (the natural notion of equivalence for topological spaces).
To be more precise, you could make the following definition:
Definition: Let $P$ be a property of a topological space $X$. We say that $P$ is a topological property if given any topological space $Y$ and a homeomorphism $f:X\to Y$, then $Y$ has property $P$ as well.
Then, the phrase you've written can be rewritten as follows:
Proposition: Compactness and connectedness are both topological properties.
Alex Mathers
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1(Though $P$ isn't an object of ZFC - either $P$ is something like a definable proper class, or you do something like assume the Axiom of Universes and then define $P$ to be a subset of the "small" topological spaces.) – Daniel Schepler Oct 26 '17 at 22:47
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@DanielSchepler: so what? Who mentioned ZFC? It is very well known that set theory provides at best an awkward foundation for much of mathematics. The OP is asking a good question about concepts that ZFC can't model well. There are lots of ways of addressing the shortcomings of ZFC. – Rob Arthan Oct 26 '17 at 23:01
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Getting off topic - I tend to agree, and in fact I tend to like type-theory based foundations as better reflections of how everyday mathematics gets done. But even there, there are similar issues in many type-theoretic foundational systems such as the issue that forces Coq to introduce the
Typeuniverse hierarchy. (Even though this question wouldn't run afoul of the issue.) So my point was just that depending on your foundations, there might be some subtleties involved in formalizing exactly kind of object $P$ is. – Daniel Schepler Oct 26 '17 at 23:08
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According to the Handbook of mathematics (see p. 197), it means that compactness and connectedness are Topological invariants.
Pedro
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