MATH1373
... Q3: Let A be a subset of a topological space X, prove that A is closed in X if and only if A contains all of its limit points. Q4: Prove: If A is any subset of a closed set B, then A B . Q5: Show by a counter example that the function f which assigns to each subset A its interior, i.e, f A I ...
... Q3: Let A be a subset of a topological space X, prove that A is closed in X if and only if A contains all of its limit points. Q4: Prove: If A is any subset of a closed set B, then A B . Q5: Show by a counter example that the function f which assigns to each subset A its interior, i.e, f A I ...
An Introduction to Functions
... Why you should learn it To represent real-life relationships between two quantities such as time and altitude for a rising hot-air balloon. ...
... Why you should learn it To represent real-life relationships between two quantities such as time and altitude for a rising hot-air balloon. ...
1. For ƒ(x)
... The value of the function for an input of 24 is 6.48. This means that it costs $6.48 to develop 24 photos. ...
... The value of the function for an input of 24 is 6.48. This means that it costs $6.48 to develop 24 photos. ...
Problem Sheet 2 Solutions
... Suppose X is Hausdorff and let x, y ∈ X with x 6= y. Then there exist open sets U , V with x ∈ U , y ∈ V and U ∩ V = ∅. Since U and V are open and non-empty, we see that X \ U and X \ V are both finite and hence the union (X \ U ) ∪ (X \ V ) is finite. But (X \ U ) ∪ (X \ V ) = X \ (U ∩ V ) = X \ ∅ ...
... Suppose X is Hausdorff and let x, y ∈ X with x 6= y. Then there exist open sets U , V with x ∈ U , y ∈ V and U ∩ V = ∅. Since U and V are open and non-empty, we see that X \ U and X \ V are both finite and hence the union (X \ U ) ∪ (X \ V ) is finite. But (X \ U ) ∪ (X \ V ) = X \ (U ∩ V ) = X \ ∅ ...
Chapter Two
... The most basic use of limits is to describe how a function behavior as the independent variable approaches a given value. DEFINITION: If the value of f(x) can be made as close as we like to L by taking the value of x sufficiently close to a (but not equal a), then we write: ...
... The most basic use of limits is to describe how a function behavior as the independent variable approaches a given value. DEFINITION: If the value of f(x) can be made as close as we like to L by taking the value of x sufficiently close to a (but not equal a), then we write: ...
Definition of a Topological Space Examples Definitions Results
... is an accumulation point or limit point of A if every open set of X that contains p also contains a point of A distinct from p. Def. Let A ⊂ X , where X is a topological space. The derived set of A, A , is the set of all accumulation points of A. Theorem: A = A ∪ A . Note: This implies A ⊂ A . ...
... is an accumulation point or limit point of A if every open set of X that contains p also contains a point of A distinct from p. Def. Let A ⊂ X , where X is a topological space. The derived set of A, A , is the set of all accumulation points of A. Theorem: A = A ∪ A . Note: This implies A ⊂ A . ...
