Let \(X\) and \(Y\) be topological spaces and \(f: X \to Y\) be a continuous function. If \(A\) is a subspace of \(X\text{,}\) prove that \(f|_A : A \to Y\) is also continuous.
2.
Let \(X\) be a topological space, let \(A\) be a subspace of \(X\text{,}\) and let \(B\) be a subspace of \(A\text{.}\) Show that the subspace topology that \(B\) inherits from \(A\) is the same as the subspace topology that \(B\) inherits from \(X\text{.}\)
3.
Let \(A\) be a subspace of a topological space \(X\) and let \(B\) be a subset of \(A\text{.}\)
(a)
Prove that a point \(x\) in \(A\) is a limit point of \(B\) in the subspace topology for \(A\) if and only if \(x\) is a limit point of \(B\) in the topology on \(X\text{.}\)
(b)
Prove that the closure of \(B\) in the subspace topology for \(A\) is equal to \(\overline{B} \cap A\text{,}\) where \(\overline{B}\) is the closure of \(B\) in \(X\text{.}\)
4.
Show that \(\R\) is homeomorphic, with the standard topology, to any interval of the form \((a,\infty)\) or \((-\infty,b)\text{.}\)
5.
Let \(X\) be a topological space.
(a)
Let \(O\) be an open subset of \(X\text{.}\) Prove that a subset \(A\) of \(O\) is open in \(O\) if and only if \(A\) is open in \(X\text{.}\)
(b)
Let \(C\) be a closed subset of \(X\text{.}\) Prove that a subset \(B\) of \(C\) is closed in \(C\) if and only if \(B\) is closed in \(X\text{.}\)
6.
A property of a topological space is said to be hereditary if that property is inherited by every subspace. We state this more formally in the following definition.
Definition15.3.
A property \(P\) of a topological space \(X\) is hereditary if every subspace of \(X\) also has property \(P\text{.}\)
Show that properties \(T_1\text{,}\)\(T_2\text{,}\) and \(T_3\) are hereditary. (The separation axioms \(T_i\) are found in Chapter 13.) The fact that \(T_4\) is not hereditary is somewhat difficult. One example is the Tychonoff plank (which is normal) with the Deleted Tychonoff plank (which is not normal) as subspace. An interested reader can consult Counterexamples in Topology (2nd ed.), Lynn Arthur Steen and J. Arthur Seebach, Jr., Dover Publications, 1978.
7.
Suppose that \(f : X \to Y\) is a homeomorphism from a topological space \(X\) to a topological space \(Y\text{.}\) Let \(a \in X\text{.}\) Must the subspace \(X' = X \setminus \{a\}\) of \(X\) be homeomorphic to the subspace \(Y' = Y \setminus \{f(a)\}\) of \(Y\text{?}\) Prove your conjecture.
8.
For each of the following, answer true if the statement is always true. If the statement is only sometimes true or never true, answer false and provide a concrete example to illustrate that the statement is false. If a statement is true, explain why.
(a)
If \(X\) has the discrete topology, then every subspace of \(X\) has the discrete topology.
(b)
If \(X\) is a topological space that does not have the discrete topology, then no subspace of \(X\) has the discrete topology.
(c)
If \(f : X \to Y\) is a continuous function between topological spaces \(X\) and \(Y\text{,}\) and \(X\) is Hausdorff, then the subspace \(f(X)\) of \(Y\) is Hausdorff.
(d)
If \(A\) is a subspace of a topological space \(X\) and \(B\) is a subset of \(X\text{,}\) then the closure of \(B \cap A\) in the subspace topology for \(A\) equals \(\overline{B} \cap A\text{,}\) where \(\overline{B}\) is the closure of \(B\) in \(X\text{.}\)
(e)
If \(A\) is a subspace of a topological space \(X\) and \(C\) is a subset of \(X\text{,}\) then the interior of \(C \cap A\) in the subspace topology for \(A\) equals \(\Int(C) \cap A\text{,}\) where \(\Int(C)\) is the interior of \(C\) in \(X\text{.}\)
