Chapter 4

If \(\operatorname{card}(X) \geq 2\), there is a topology on \(X\) that is \(T_{0}\) but not \(T_{1}\).

If \(X\) is an infinite set, the cofinite topology on \(X\) is \(T_{1}\) but not \(T_{2}\), and is first countable iff \(X\) is countable.

Every metric space is normal. (If \(A, B\) are closed sets in the metric space \((X, \rho)\). consider the sets of points \(x\) where \(\rho(x, A)<\rho(x, B)\) or \(\rho(x, A)>\rho(x, B) .)\)

Every separable metric space is second countable.

If \(X\) is an infinite set with the cofinite topology and \(\left\\{x_{j}\right\\}\) is a sequence of distinct points in \(X\), then \(x_{j} \rightarrow x\) for every \(x \in X\).

If \(X\) is a linearly ordered set, the topology \(\mathcal{T}\) generated by the sets \(\\{x: xa\\}(a \in X)\) is called the order topology. a. If \(a, b \in X\) and \(a

A topological space \(X\) is called disconnected if there exist nonempty open sets \(U, V\) such that \(U \cap V=\varnothing\) and \(U \cup V=X\); otherwise \(X\) is connected. When we speak of connected or disconnected subsets of \(X\), we refer to the relative topology on them. a. \(X\) is connected iff \(\varnothing\) and \(X\) are the only subsets of \(X\) that are both open and closed. b. If \(\left\\{E_{\alpha}\right\\}_{\alpha \in A}\) is a collection of connected subsets of \(X\) such that \(\bigcap_{\alpha \in A} E_{\alpha} \neq\) \(\varnothing\), then \(\bigcup_{\alpha \in A} E_{\alpha}\) is connected. c. If \(A \subset X\) is connected, then \(\bar{A}\) is connected. d. Every point \(x \in X\) is contained in a unique maximal connected subset of \(X\), and this subset is closed. (It is called the connected component of \(x\).)

If \(X\) is a topological space, \(A \subset X\) is closed, and \(g \in C(A)\) satisfies \(g=0\) on \(\partial A\), then the extension of \(g\) to \(X\) defined by \(g(x)=0\) for \(x \in A^{c}\) is continuous.

Let \(X\) be a topological space, \(Y\) a Hausdorff space, and \(f, g\) continuous maps from \(X\) to \(Y\). a. \(\\{x: f(x)=g(x)\\}\) is closed. b. If \(f=g\) on a dense subset of \(X\), then \(f=g\) on all of \(X\).

If \(X\) and \(Y\) are topological spaces and \(y_{0} \in Y\), then \(X\) is homeomorphic to \(X \times\left\\{y_{0}\right\\}\) where the latter has the relative topology as a subset of \(X \times Y\).