## 5.2 Basic notions

The following is a list of basic notions in topology. Some of these notions are discussed in more detail in the text that follows and some are defined in the list, but others are considered basic and will not be defined. If you are not familiar with most of the italicized concepts, then we suggest looking at an introductory text on topology before continuing.

$X$ is a

*topological space*,$x\in X$ is a

*point*,$E \subset X$ is a

*locally closed*subset,$x\in X$ is a

*closed point*,$E \subset X$ is a

*dense*subset,$f : X_1 \to X_2$ is

*continuous*,an extended real function $f : X \to \mathbf{R} \cup \{ \infty , -\infty \} $ is

*upper semi-continuous*if $\{ x \in X \mid f(x) < a\} $ is open for all $a \in \mathbf{R}$,an extended real function $f : X \to \mathbf{R} \cup \{ \infty , -\infty \} $ is

*lower semi-continuous*if $\{ x \in X \mid f(x) > a\} $ is open for all $a \in \mathbf{R}$,a continuous map of spaces $f : X \to Y$ is

*open*if $f(U)$ is open in $Y$ for $U \subset X$ open,a continuous map of spaces $f : X \to Y$ is

*closed*if $f(Z)$ is closed in $Y$ for $Z \subset X$ closed,a

*neighbourhood of $x \in X$*is any subset $E \subset X$ which contains an open subset that contains $x$,the

*induced topology*on a subset $E \subset X$,$\mathcal{U} : U = \bigcup _{i \in I} U_ i$ is an

*open covering of*$U$ (note: we allow any $U_ i$ to be empty and we even allow, in case $U$ is empty, the empty set for $I$),a

*subcover*of a covering as in (13) is an open covering $\mathcal{U}' : U = \bigcup _{i \in I'} U_ i$ where $I' \subset I$,the open covering $\mathcal{V}$ is a

*refinement*of the open covering $\mathcal{U}$ (if $\mathcal{V} : U = \bigcup _{j \in J} V_ j$ and $\mathcal{U} : U = \bigcup _{i \in I} U_ i$ this means each $V_ j$ is completely contained in one of the $U_ i$),*$\{ E_ i \} _{i \in I}$ is a fundamental system of neighbourhoods of $x$ in $X$*,a topological space $X$ is called

*Hausdorff*or*separated*if and only if for every distinct pair of points $x, y \in X$ there exist disjoint opens $U, V \subset X$ such that $x \in U$, $y \in V$,the

*product*of two topological spaces,the

*fibre product $X \times _ Y Z$*of a pair of continuous maps $f : X \to Y$ and $g : Z \to Y$,the

*discrete topology*and the*indiscrete topology*on a set,etc.

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