How do you prove T is topology?

Basic proof that T is a topology.

1. The empty set and X are in T.
2. The intersection of any finite collection of subsets of X in T is also in T.
3. The union of any collection of sets in T is also in T.

How many topologies are possible on a set of 2 points?

three
So, in fact, there are only three inequivalent topologies on a two-point set: the trivial one, the discrete one, and the Sierpiński topology.

Is Z homeomorphic to Q?

No, since Z is discrete, but Q is not. Suppose that there exists a homeomorphism f:Q→Z, f−1(0) is open since {0} is open in Z, any non empty open subset of Q contains more than one element.

Is trivial topology hausdorff?

The trivial topology is the topology with the least possible number of open sets, namely the empty set and the entire space, since the definition of a topology requires these two sets to be open. In particular, it is not a Hausdorff space. Not being Hausdorff, X is not an order topology, nor is it metrizable.

What is not a topology?

Given the set of integers, the family of all finite subsets of the integers plus itself is not a topology, because (for example) the union of all finite sets not containing zero is not finite but is also not all of. and so it cannot be in.

Which of the following is not an example of topology?

(d) Connect is the right answer. Explanation: The types of topology are bus topology, ring topology, star topology, mesh topology and hybrid topology. Connect is not one of them.

Is r2 and r3 homeomorphic?

But minus a point is disconnected and minus a point is connected, and a disconnected space cannot be homeomorphic to a connected space. …

Is Q homeomorphic to n explain?

Therefore all of the sequences in Q are mapped to a sequence in N preserving limits. But since sequences in N converge constantly, this cannot be a bijection. therefore they are not homeomorphic.