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LMKIYHAQ
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Homework Statement
Let X be a space. A[tex]\subseteq[/tex]X and U, V, W [tex]\in[/tex] topolgy(X). If W[tex]\subseteq[/tex] U[tex]\cup[/tex] V and U[tex]\cap[/tex] V[tex]\neq[/tex] emptyset,
Prove bd(W) = bd(W[tex]\cap[/tex]U) [tex]\cup[/tex] bd (W[tex]\cap[/tex] V)
Homework Equations
bd(W) is the boundary of W...
I think I have the "[tex]\supseteq[/tex]" part, but I am having trouble with the "[tex]\subseteq[/tex]" part.
The Attempt at a Solution
[tex]\supseteq[/tex]: Assume x [tex]\in[/tex] bd(W[tex]\cap[/tex]U) [tex]\cup[/tex] bd(W [tex]\cap[/tex]V). Show x[tex]\in[/tex] bd(W). Then x [tex]\in[/tex] bd(W[tex]\cap[/tex] U) or x[tex]\in[/tex] bd(W[tex]\cap[/tex] V). If x[tex]\in[/tex] bd (W[tex]\cap[/tex]U) means x is in bd(W) since W[tex]\cap[/tex]U[tex]\neq[/tex] emptyset and since W[tex]\subseteq[/tex] U[tex]\cup[/tex] V, some part of W[tex]\subseteq[/tex]U.
If x[tex]\in[/tex] bd(W[tex]\cap[/tex] V) then x[tex]\in[/tex] bd(W) since W[tex]\cap[/tex] V[tex]\neq[/tex] emptyset and since W[tex]\subseteq[/tex]U[tex]\cap[/tex] V, some part of W[tex]\subseteq[/tex] V.
Does this look ok for this part of the proof?
[tex]\subseteq[/tex]: Assume x[tex]\in[/tex]bd(W). Show x[tex]\in[/tex] bd(W[tex]\cap[/tex] U)[tex]\cup[/tex] (bd(W[tex]\cap[/tex] V). Since U and V are disjoint and W[tex]\subseteq[/tex]U[tex]\cup[/tex], W_U[tex]\subseteq[/tex]U or W_V[tex]\subseteq[/tex]V. Suppose W_U[tex]\subseteq[/tex]U then x[tex]\in[/tex]bd(W_U)[tex]\subseteq[/tex]bd(W_U[tex]\cap[/tex]U).Suppose W_V[tex]\subseteq[/tex] V then x[tex]\in[/tex] bd(W_V)[tex]\subseteq[/tex]bd(W_V[tex]\cap[/tex] V). Since x[tex]\in[/tex]bd(W)[tex]\subseteq[/tex]bd(W_U[tex]\cap[/tex]U) or bd(W_V[tex]\cap[/tex] V) then x[tex]\in[/tex] bd(W[tex]\cap[/tex] U)[tex]\cup[/tex] (bd(W[tex]\cap[/tex] V).
I am not sure if I need to specify what W_U and W_V are? or if this even works for this second part of the proof?
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