Re: Topology with subspace and limit point.
- From: José Carlos Santos <jcsantos@xxxxxxxx>
- Date: Mon, 05 Nov 2007 14:14:11 +0000
On 05-11-2007 12:19, mina_world wrote:
If Y is a subspace of X, and B is a subset of Y,
Show that x in B' in Y <=> x in B' in X.
(x in Y)
--------------------------------------------------
(=>)
Let (X, T_x) , (Y, T_y).
let x not in B' in X.
There is open V_x in T_x such that (V_x) /\ (B-{x}) = empty.
(x in V_x)
so, [(V_x) /\ Y] /\ (B-{x}) = empty.
Since (V_x) /\ Y is open in T_y, x not in B' in Y.
(<=)
I don't know well.
or I can't find a counter-example, too.
x in B' in X <=> every neighborhood V of x in X intersects B
=> every neighborhood V of x in Y intersects B
because V = W /\ Y for some neighborhood W of x in X, W intersects B
(by assumption) and every element of B is also an element of Y.
Best regards,
Jose Carlos Santos
.
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