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.

1. $X$ is a topological space,

2. $x\in X$ is a point,

3. $E \subset X$ is a locally closed subset,

4. $x\in X$ is a closed point,

5. $E \subset X$ is a dense subset,

6. $f : X_1 \to X_2$ is continuous,

7. 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 $a \in \mathbf{R}$,

8. 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 $a \in \mathbf{R}$,

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

10. a continuous map of spaces $f : X \to Y$ is closed if $f(Z)$ is closed in $Y$ for $Z \subset X$ closed,

11. a neighbourhood of $x \in X$ is any subset $E \subset X$ which contains an open subset that contains $x$,

12. the induced topology on a subset $E \subset X$,

13. $\mathcal{U} : U = \bigcup _{i \in I} U_ i$ 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 $I$),

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

15. 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 $U_ i$),

16. $\{ E_ i \} _{i \in I}$ is a fundamental system of neighbourhoods of $x$ in $X$,

17. 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 $x \in U$, $y \in V$,

18. the product of two topological spaces,

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

20. the discrete topology and the indiscrete topology on a set,

21. etc.

Comment #1469 by David Whitten on

Notion 2 and 4 appear to be identical. Since E is defined in 3, shouldn't 4 read x is an element of E ? There is an undefined free variable I in 11, 12, and 13. Is there an unstated assumption that it is the set of integers ? Or that it is a subset of U ?

Comment #1470 by sdf on

These are not definitions, which is the key point regarding 2 and 4. The different instances of E are not related. It is implicit in 11,12,13 that I is some arbitrary set finite or infinite usually just called an indexing set.

Comment #1471 by sdf on

Actually I see now where the confusion arises. Despite what is in the first line of the section, no. 7,8,9,12 are in fact definitions, but the others are not.

Comment #7729 by Giacomo on

I think the language here is confusing, what if change it into

(1) a topological space $X$, (2) a point $x \in X$, (3) a locally closed subset $E \subset X$, ...

Comment #8640 by Matthieu Romagny on

I am not familiar with the notation $\mathcal{U}:U$, which is not defined here. What does it mean precisely? Is it standard?

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