Suppose (X,T) is a topological space. Let Y be non-empty subset of X. The the set J={intersection(Y,U) : U is in T} is called the subspace toplogy on Y. Prove that J indeed a toplogy on Y i.e., (Y,J) is a topological space.

Solution Summary

Topological subspaces are investigated. The solution is detailed and well presented.

19. Let X be a topological space and let Y be a subset of X. Check that the so-called subspacetopology is indeed a topology of Y.
(question is also included in attachment)

Consider the following subsets of (FUNCTION1) and (FUNCTION2). The subspaces X and Y of (SYMBOL) inherit the subspacetopology. 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, pleas

2. Let X be given the co-finite topology. Suppose that X is a infinite set. Is X Hausdorff? Should compact subsets of X be closed? What are compact subsets of X?
3. Let (X,T) be a co-countable topological space. Show that X is connected if it is uncountable. In fact, show that every uncountable subspace of X is connected.

Let Y be a subspace of X and let A be a subset of Y. Denote by Cl(A_X) the closure of A in the topological space X and by Cl(A_Y) the closure of A in the topological space Y. Prove that Cl(A_Y) is a subset of Cl(A_X) . Show that in general Cl(A_Y) not equal Cl(A_X). See the attached file.

Calculate the dimension of the subsapce of P4(t) (polynomials of degree at most 4 in the indeterminate t) consisting of polynomials p(t) E P4(t) satisfying p(0) = p(1)
Please show complete steps to this vector subspace dimensions question.

Let X = Y = IR and f: X -> Y be given by
f(x) = { x^2 + 1 x >= 0
{ 0 x < 0
Consider the following statement
A is open in Y then f^-1(A) is open in X.
With the following settings on X,Y, determine whether the above statement is true
a) X,Y are given the usual topology

Question: Find the interior, the closure, the accumulation points, the isolated point and the boundary points of the following sets.
a) X = [(0,1) in R with the topology induced by d(x,y) = |x-y|]
b) X = Q in R with the same topology as above.
c) X = {(x,y) : |y| < x^2} U {(0,y), y E R} in R^2, with the topology induced by

Let Y be a subspace of X and let A be a subset of Y. Denote by Int_X(A) the interior of A in the topological space X and by Int_Y(A) the interior of A in the topological space Y. Prove that Int_X(A) is a subset of Int_Y(A). Illustrate by an example the fact that in general Int_X(A) not Int_Y(A).