Given a topological space \((X, \tau)\) and a subset \(Y \subseteq X\), we can define the subspace topology \(\tau_Y\) on \(Y\) to be the topology on \(Y\) consisting of the open sets \(U \cap Y\) where \(U\) is an arbitrary \(\tau\)-open subset of \(X\).

We checked that this is indeed a topology and also checked that if we have a topological basis on \(X\) then in a similar way we can get a natural tolological basis on \(Y\).

Given the topological space \(\mathbb{R}\) with the standard topology and \(Y = [0, 1]\) we asked if there is a “nice” basis for the subspace topology on \(Y\). We got the sets \(\{(a, b) \mid 0 \le a < b \le 1\} \cup \{[0, b) \mid 0 < b \le 1\} \cup \{(a, 1] \mid 0 \le a < 1\} \cup \{\emptyset\}.\)

Given the topological space \(\mathbb{R}\) with the standard topology and \(Y = [0, 1) \cup \{2\}\) we asked if there is a “nice” basis for the subspace topology on \(Y\). We got the sets \(\{(a, b) \mid 0 \le a < b \le 1\} \cup \{[0, b) \mid 0 < b \le 1\} \cup \{2\} \cup \{\emptyset\}.\)

Given topological spaces \((X, \tau_X)\) and \((Y, \tau_Y)\) we defined the product topology on \(X \times Y\) to be the topology \(\tau_{X \times Y}\) on \(X \times Y\) generated by the basis of sets of the form \(U \times V\) where \(U\) and \(V\) are arbitrary \(\tau_X\)-open and \(\tau_Y\)-open sets respectively.

We checked that these sets do indeed form a topological basis and thus generate a topology.

We sketched an informal proof (with pictures) that \(\mathbb{R}^2\) along with the topology generated by the standard Euclidean metric \(d((x_1, y_1), (x_2, y_2)) = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\) is the same topology as the product topology of \(\mathbb{R}\) with the standard topology with itself. This shows that there are two very different descriptions of the same topology on \(\mathbb{R}^2\).

Finally, we introduced the notion of functions between topological spaces that respect the underlying structure. Given topological spaces \((X, \tau_X)\) and \((Y, \tau_Y)\), a function \(f : X \rightarrow Y\) is continuous if for any \(\tau_Y\)-open \(U \subseteq Y\) the preimage \(f^{-1}(U)\) is \(\tau_X\)-open.

We checked that the following functions are indeed continuous:

  1. The function \(f : \mathbb{R} \rightarrow \mathbb{R}\) defined by \(f(x) = 2x + 1\) is continuous with the respect the the standard topology on \(\mathbb{R}\) on both the domain and codomain
  2. The function \(\pi_1 : \mathbb{R}^2 \rightarrow \mathbb{R}\) defined by \(\pi_1(x, y) = x\) is continuous with respect to the standard topology on \(\mathbb{R}\) and the standard topology on \(\mathbb{R}\)
  3. The function \(f : \mathbb{R} \rightarrow \mathbb{R}\) defined by \(f(x) = 2\) is continuous with the standard topology on \(\mathbb{R}\) on both the domain and the codomain.