Topology: An Inquiry-Based Approach

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.

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 $X\text{.}$

Show that $\mathbb{R}$ is homeomorphic, with the standard topology, to any interval of the form $(a,\mathrm{\infty })$ or $(-\mathrm{\infty },b)\text{.}$

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.

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.

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 $X^{\prime }=X\setminus \{a\}$ of $X$ be homeomorphic to the subspace $Y^{\prime }=Y\setminus \{f(a)\}$ of $Y\text{?}$ Prove your conjecture.