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Munkres Topology Ch. 1 section 2 Ex #1
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[QUOTE="Infrared, post: 6355873, member: 467682"] This is basically correct, but not what you did in your OP. The LHS is a little clunky though- why would you write ##\{x: x\in E\}## instead of just ##E.##? In general though, when writing sets in this way, it's better to write something like ##\{x\in C: \Phi(x)\}## instead of ##\{x:\Phi(x)\}## where ##\Phi(x)## is some predicate. If you don't include a domain, then you can get issues like Russel's paradox (thinking about the "set" ##\{x:x\notin x\}## leads to contradictions) as well as just annoying ambiguities, e.g. in ##\{x: x^3=1\}##, is ##x## supposed to be a real number, or a complex number, or maybe a matrix...? Here it's clear what you mean, but just be careful. If ##f## is invertible, then ##f^{-1}(b)## makes sense- it is just the function ##f^{-1}## applied to the element ##b.## If ##f## is not necessarily invertible, then to be perfectly strict, you should only write ##f^{-1}(C)## where ##C## is a subset of ##B##, but sometimes ##f^{-1}(b)## is written as shorthand for ##f^{-1}(\{b\})##, and this latter expression is well-defined (it's the set of all elements of ##A## that map to ##b##). For example, if ##f:\mathbb{R}\to\mathbb{R}## is the function ##f(x)=x^2##, then ##f^{-1}(1)=f^{-1}(\{1\})=\{1,-1\}## according to this notation. ##f^{-1}## is not a function on its own. It might be a good idea to avoid this shorthand if you're still getting used to this stuff. [/QUOTE]
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Munkres Topology Ch. 1 section 2 Ex #1
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