Problem: ( f: \mathbbR \to \mathbbR ), usual topology. Show ( f(x) = 2x ) is continuous at ( x=2 ).
Tutor outline:
Common Query: "Prove that ( f: X \to Y ) is continuous if and only if for every ( x \in X ) and every neighborhood ( N ) of ( f(x) ), there is a neighborhood ( M ) of ( x ) such that ( f(M) \subset N )."
Why it’s hard: This is the topological rephrasing of the epsilon-delta definition. Students often confuse the direction of the mapping. A robust solution set will restate the definition of a neighborhood (an open set containing the point) and show how the "pre-image of open is open" condition is equivalent to the local condition.
Problem: Show that the discrete metric ( d(x,y) = 0 ) if ( x=y ), else 1, induces the discrete topology. Introduction To Topology Mendelson Solutions
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Problem: In a metric space, prove closure of ( E ) is closed.
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Problem: Prove that closed subset of compact space is compact. Problem: ( f: \mathbbR \to \mathbbR ), usual topology
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Problem: Show that compact subset of Hausdorff space is closed.
Solution:
Problem (paraphrased):
Let ( X = a,b,c ) with topology ( \tau = \emptyset, a, b, a,b, X ). Is ( c ) closed? Common Query: "Prove that ( f: X \to
Solution outline (tutor view):
✔ Check: “Closed does not mean ‘not open’ – here ( c ) is not open, but that’s irrelevant.”
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