A probability space consists of a sample space, X, which is a set of all possible random elements. A collection of random events which is a sigma field, F, which consists of all the events that belong to X. A probability measure that assigns a number between [0,1] for each event in the sigma field F and statisfies the axioms of probability.
A topological space is a set, X, such that a topology, T, has been specified (under the 3 main topological properties).
I read that: Topological spaces are used to define a notion of "closeness". With it, you can intuitively speak about points which are close to each other. (However, we may not know how close: this is a metric space). …. A measure space serves an entirely different goal. A measure space is made to define integrals… Reference https://www.physicsforums.com/threads/measurable-spaces-vs-topological-spaces.558644/
A metric space is a metrizable space X with a specific metric d that gives the topology of X. Therefore, there is a connection between a metric space and a topological space namely a topological space may be induced by a metric space. Is that true? In this case the connection between topological spaces and measureable spaces can be changed to be: What is the relationship between measurable spaces and metric spaces a question that has some relevant answer in What's the relationship between a measure space and a metric space?
But I still do not see a well-written answer to that question and how is it connected to topological spaces with a defined metrics? Any resources or help is appreciated Thank you very much in advance