1

A topological space is called a US-space provided that each convergent sequence has a unique limit.

Each Fréchet $US$-space $X$ is a $KC$-space.

Proof. Suppose that $x \in K$ where $K$ is a compact subspace of $X$. Because X is a Fréchet space, there is a sequence $(k_n)_{n∈N}$ of points of $K$ converging to $x$. Since $K$ is compact, that sequence has a cluster point $c$ in $K$. There is a subsequence of $(k_n)_{n∈N}$ converging to $c$ . Hence $x = c \in K$, because $X$ is a $US$-space. so $K$ is closed and then $X$ is a $KC$-space.

Why "there is a subsequence of $(k_n)_{n∈N}$ converging to $c$"? How does it build?

Stefan Hamcke
  • 28,621
fatemeh
  • 427

1 Answers1

1

A US space is also $T_1$ since if $x$ is a point with another point $y$ in its closure then the constant sequence at $x$ converges to both $x$ and $y$, so by uniqueness of limits $x$ must equal $y.$ So points are closed.

Now, in a Fréchet-Urysohn space with the $T_1$-property a cluster point of a sequence is also a limit point of some subsequence (see my answer to your previous question).

Community
  • 1
Stefan Hamcke
  • 28,621
  • Is $T_ 1$ -property necessary? which book is your references for my previous question? – fatemeh Sep 20 '13 at 14:10
  • I don't know if it's necessary, but it is sufficient. And I don't have a book as a reference. – Stefan Hamcke Sep 20 '13 at 14:16
  • What is exactly your definition of cluster point, here? I thought $x \in X$ is a cluster poinf of sequence ${x_n}$ in $( X,\tau)$ if for $ k \in ‎‎\mathbb{N}$ and open $G$ that contains $x$ ,$ G \cap { x_n : n\geq k } \not= \emptyset$.is it right? – fatemeh Sep 20 '13 at 16:44
  • @fatemeh: That's right. – Stefan Hamcke Sep 20 '13 at 16:49