abstract analysis

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Abstract Analysis

Show that phi is (infinite d,d) continuous where d is the standard metric... (See attachment for full question)

Continuity and Limit Points

1. For i = 1,2 let fi: Xi --> Yi be maps between topological spaces. Show that the product f1Xf2: X1XX2 --> Y1XY2 defined by f1Xf2(x1x2):= (f1(x1), f2(x2)) is continuous if and only if f1 and f2 are continuous. *(Please see attachment for proper representation of formulas and problem #2)

Subspace Topology: Interior, Closure, Boundary and Limit Points

Consider the following subsets of (FUNCTION1) and (FUNCTION2). The subspaces X and Y of (SYMBOL) inherit the subspace topology. In the following cases determine the interior, the closure, the boundary and the limit points of the subsets: 1, 2 and 3 *(For complete problem, including properly cited functions and symbols, please see attachment)

Limit Points (Continuous Map)

Let f: X --> Y be a continuous map. Let A (SYMBOL) C. Show that, if (FUNCTION1) is closed, then (FUNCTION2). *(For complete problem, including proper citation of functions and symbols, please see attachment)