Relation between half smash and smash product

Question

Let $X$ be a pointed space with base point $x_0$ and $B$ be an unpointed space. Define the half smash product between $X$ and $B$ mas follows:

$$B \ltimes X:= \frac{B \times X}{B \times \{x_0\}}$$

Now consider space $B_+ = B \coprod \{+\}$, ie $B$ with an additional point $+$. So, we can think $B_+$ as a pointed space with base point $+.$ Then the smash product is $B_+ \wedge X = \frac{B_+ \times X}{B_+ \vee X}$

Statement: The spaces $B \ltimes X$ and $B_+ \wedge X$ are homotopic.

Is the above statement is true? If true can you suggest any reference?

Any help will be appreciated.

Thank you in advance.

Answer 1

The spaces $B_+ \wedge X$ and $B \rtimes X$ are the same space... If you want to be very pedantic they are homeomorphic.

$B_+ \times X = B \times X \sqcup \{pt\} \times X$, and you are collapsing $B_+ \vee X = B \times \{pt\} \cup \{pt\} \times X$ to a point. That gives us precisely $(B \times X)/(B \times pt)$