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I am new student in the course of topology so i am confused whether every set is a topological space ? If anyone could help me out for this I will remain thankful.

abstract
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    There has to be a collection of subsets called "open" for a topology. – coffeemath Jan 14 '18 at 08:00
  • On any nonempty set, one can define a topology by specifying a collection of sets which satisfy the axioms for open sets. If the set has more than one point, there will be more than one such topology. Thus, a set by itself is not a topological space until you declare a choice of topology. However, for some sets, there is a default choice of topology, which is assumed unless an alternate topology is specified. – quasi Jan 14 '18 at 08:05

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As coffeemath states, a topological space is a set equipped with a collection of subsets of that set, called the open subsets, that satisfy certain laws. So a set by itself is not a topological space. That said, every set gives rise to a topological space. In particular, it gives rise to the discrete topological space where we take the set of all subsets, i.e. the powerset, as the set of open subsets. This is the most natural way to view an arbitrary set as a topological space. There are other topological structures that you could induce given an arbitrary set. For example, the indiscrete topology which takes only the empty set and the set itself as open subsets. However, in some sense, this isn't too different from collapsing the set to a single point (assuming it is non-empty).

It's extremely common in mathematics in general to talk about a notion of "structured set" while leaving the "structure" on the set implicit in examples when it is deemed "obvious" or "implied". So, for example, you'll often see $\mathbb{R}$, the set of real numbers, being called a topological space by which is meant $\mathbb{R}$ equipped with the set of arbitrary unions of open intervals as the open sets (i.e. the topology induced by the metric structure on $\mathbb{R}$). There are many other possibilities (such as the discrete or indiscrete topologies, but many others besides those).

Derek Elkins left SE
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  • If I say " X " is a topological space , what does it mean ? does it mean that there is at least one topology on X ? – abstract Jan 14 '18 at 08:12
  • Yes. It means that you have a pair $(X,\mathcal{T})$ with $\mathcal{T}$ a collection of subsets satisying the axioms in the answer below. – B. Pasternak Jan 14 '18 at 08:14
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    Technically it doesn't mean anything, but among educated mathematicians nobody would say that unless there's an obvious/standard topology to consider. – Henrik supports the community Jan 14 '18 at 08:16
  • @AbdulFatahRajri Every set forms a topological space in at least two ways via the discrete and indiscrete topologies (except for the empty set and singleton sets where those coincide), so the mere existence of a topology is tautological. It is common for that the topology is not explicitly named, as in B. Pasternak's comment, because it can be referred to indirectly, e.g. "Let $X$ and $Y$ be topological spaces and $f:X\to Y$ a continuous function, then $f^{-1}(U)$ is open in $X$ for any open subset $U$ of $Y$." Something like $\mathcal{O}(X)$, may be used to name the set of open sets of $X$. – Derek Elkins left SE Jan 14 '18 at 08:33
  • And Is there an example except null set over which we can't define topology? – abstract Jan 14 '18 at 08:51
  • @AbdulFatahRajri The empty set does have (exactly one) topology. Every set can be given a topology. There are no counter-examples. You can always use the discrete topology, and you can always use the indiscrete topology. In some cases, these are the same topology, but usually they are not. In particular, for any set containing at least two distinct elements, they will not be the same topology. – Derek Elkins left SE Jan 14 '18 at 09:19
  • Thank you very much for responding – abstract Jan 14 '18 at 09:25
  • I strongly agree with @Henrik, for the most common spaces we know/consider there is a somewhat "obvious" topology. Nevertheless we can define more than one topology $\tau$ on a set $X$ (in most cases) that can lead to very different topological spaces $(X,\tau)$ (see for instance https://en.wikipedia.org/wiki/Lower_limit_topology). – noctusraid Jan 14 '18 at 12:02