# How to determine if a set is open or closed

Let \$f: sarkariresultonline.infobbR o sarkariresultonline.infobbR\$ be characterized by \$f (t) = t^2 \$ and also let \$U\$ be any kind of nonempty open subset of \$sarkariresultonline.infobbR\$ Then

a .\$ f (U) \$ is open

b. \$f (U) \$ is closed

c. \$f^-1(U)\$ is open up

d. \$f^-1(U)\$ is closed

Attempt is that I take any set \$U\$ consisting of facets say \$1,2,3\$ then \$f (U)\$ will have actually \$1 ,4,9\$ .Now limit suggest of \$f (U)\$ is \$phi \$ and also hence it is closed but answer appears to be \$c\$.

You watching: How to determine if a set is open or closed

Can anyone aid where I went wrong?

Hint: If \$f\$ is consistent then \$f^-1(U)\$ is open to any type of \$U subset sarkariresultonline.infobbR\$ open up set.

Edit: Consider \$f: M o N\$ consistent.

Let \$A" subset N\$ be an open up collection, we desire to present that \$f^-1(A")\$ is open. In fact, for each \$a in f^-1(A")\$, we have that \$f(a) in A"\$. By meaning of open set, there exists \$epsilon > 0\$ such that \$B(f(a), epsilon) subcollection A"\$. As \$f\$ is constant at \$a\$, there exists \$delta > 0\$ correspondent to \$epsilon > 0\$ such that

\$\$f(B(a; delta)) subset B(f(a); epsilon) subcollection A"\$\$

that is, \$\$B(a;delta) subcollection f^-1(A")\$\$

Then \$f^-1(A")\$ is open up.

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edited Dec 24 "14 at 17:13
answered Dec 24 "14 at 16:44

Aaron MarojaAaron Maroja
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Statement c is of course true, because the attribute is consistent.

If you understand that the attribute is consistent (at eextremely point), then take an open collection \$U\$. In order to present that \$f^-1(U)\$ is open, we should check out that, for \$xin f^-1(U)\$, tbelow exists \$delta>0\$ with \$\$B(x,delta)={rinsarkariresultonline.infobbR:|r-x|0\$ such that\$\$B(f(x),varepsilon)subseteq U.\$\$Now apply the \$varepsilon\$-\$delta\$ definition of continuity and the problem is solved: the required \$delta>0\$ is specifically the one provided by the continuity problem.

Why is the attribute continuous? By general theorems: the product of constant functions is continuous; the identification attribute \$tmapsto t\$ is obviously constant.

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The other statements are false, in the sense that for each one you have the right to discover a specific \$U\$ such that the condition does not organize.

For a think about \$U=sarkariresultonline.infobbR\$. Hint: \$f(U)=<0,infty)\$

For b take into consideration \$U=(0,infty)\$. Hint: \$f(U)=(0,infty)\$

For d think about \$U=(0,1)\$. Hint: \$f^-1(U)=(-1,0)cup(0,1)\$

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edited Dec 24 "14 at 18:43
answered Dec 24 "14 at 17:45

egregegreg
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