## 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.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (8)

Comment #1469 by David Whitten on

Comment #1470 by sdf on

Comment #1471 by sdf on

Comment #1489 by Johan on

Comment #7729 by Giacomo on

Comment #7979 by Stacks Project on

Comment #8640 by Matthieu Romagny on

Comment #9409 by Stacks project on