Hi! I'm trying to get some intuition for the notion of the topology of a set. The definition of a topology ##\tau## on a set ##X## is that ##\tau## satisfies the following:
- ##X## and ##Ø## are both elements of ##\tau##.
- Any union of sets in ##\tau## are also in ##\tau##.
- Any finite intersections of sets in ##\tau## are also in ##\tau##.
Alright, so one associates the set X with another set ##\tau## and the two together define a topological space if ##\tau## satisfies certain properties. Topology, as I understand it, is the study of the most qualitative features of a space. Have i understood it properly if i say that the role of the topology ##\tau## is to distribute the elements in ##X## in a certain way?
For example in the excluded point topology on the real line, the point ##p = 0##, is given by
$$\tau_E = \{\mathbb{R}\}\cup\{S \subseteq \mathbb{R} : 0 \notin S\}.$$
So while the point zero is in the set of reals it is not in the topology and therefore not in the topological space. Thus the topology has distributed the set of points in a certain way.
So how do you like intuitively to think about a topology and it's role?
- ##X## and ##Ø## are both elements of ##\tau##.
- Any union of sets in ##\tau## are also in ##\tau##.
- Any finite intersections of sets in ##\tau## are also in ##\tau##.
Alright, so one associates the set X with another set ##\tau## and the two together define a topological space if ##\tau## satisfies certain properties. Topology, as I understand it, is the study of the most qualitative features of a space. Have i understood it properly if i say that the role of the topology ##\tau## is to distribute the elements in ##X## in a certain way?
For example in the excluded point topology on the real line, the point ##p = 0##, is given by
$$\tau_E = \{\mathbb{R}\}\cup\{S \subseteq \mathbb{R} : 0 \notin S\}.$$
So while the point zero is in the set of reals it is not in the topology and therefore not in the topological space. Thus the topology has distributed the set of points in a certain way.
So how do you like intuitively to think about a topology and it's role?
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