Is hausdorff a weak topology?

Weak∗ topology on an arbitrary dual X∗ can be seen as pointwise convergence (or Tychonoff) topology (when we consider X∗ as a subset of RX, or its complex counterpart, whichever you’re more interested in), which makes it easy to show that it is Hausdorff just by noting that R (as well as C) is Hausdorff.

What do you mean by weak topology?

In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. …

Which is the weakest topology?

weak topology
Proof: The first point is clear: the norm topology already makes all linear functionals continuous. Since the weak topology is the weakest with this property, it is weaker than the strong topolgy. So every weakly open set is strongly open, and by taking complements, every weakly closed set is strongly closed.

What is weak star topology?

The weak-* (pronounced “weak star”) topology on is defined to be the -topology on , i.e., the coarsest topology (the topology with the fewest open sets) under which every element corresponds to a continuous map on . The weak-* topology is sometimes called the ultraweak topology or the. -weak topology.

When is a topological space a weak Hausdorff?

Definition 0.1. A topological space is weakly Hausdorff (or weak Hausdorff) if for any compact Hausdorff space and every continuous map, the image is closed. Every weakly Hausdorff space is (that is every point is closed), and every Hausdorff space is weakly Hausdorff. For the most common purposes for which Hausdorff spaces are used,…

Is the Hausdorff space T T 1T _ 1?

Every weakly Hausdorff space is T 1T_1 (that is every point is closed), and every Hausdorff space is weakly Hausdorff. For the most common purposes for which Hausdorff spaces are used, the assumption of being weakly Hausdorff suffices.

Which is the identity map for compact Hausdorff spaces?

In my topology lecture notes, I have written: “By considering the identity map between different spaces with the same underlying set, it follows that for a compact, Hausdorff space: $\\bullet$ any Stack Exchange Network

Which is the weak topology with respect to F?

More generally, if F is a subset of the algebraic dual space, then the initial topology of X with respect to F, denoted by σ ( X, F ), is the weak topology with respect to F . If one takes F to be the whole continuous dual space of X, then the weak topology with respect to F coincides with the weak topology defined above. for all f ∈ F and x ∈ X.